Editorial glossary
Browse existing editorial definitions and examples, then follow deeper reading in the subject’s native course. Shared senses can belong to more than one subject. Historical source definitions and tables remain in the 242-entry Physics source glossary.
808 of 808 editorial senses
68–95–99.7 rule
Statistics · Definition
For a Normal distribution, approximately 68%, 95%, and 99.7% of values lie within one, two, and three standard deviations of the mean.
The rule is an approximation for Normal or sufficiently bell-shaped data, rather than for every distribution.
For example, a Normal model with mean 50 and standard deviation 5 places about 95% of values between 40 and 60.
Absolute convergence
Calculus BC · Definition
A series converges absolutely when the series of its absolute term values converges.
Absolute convergence also guarantees convergence of the original signed series.
For example, an alternating series with magnitudes 1/n² converges absolutely.
An absolutely convergent series retains its sum under any rearrangement of its terms.
Absolute extremum
Calculus AB · Calculus BC · Definition
An absolute extremum is the greatest or least function value on the domain or interval under discussion.
The comparison includes every allowed input, including included endpoints.
For example, x² on [−1,2] has absolute maximum 4 at x=2 and absolute minimum 0 at x=0.
An absolute extremum must be attained rather than merely approached.
Absolute pressure
Physics 1 · Physics 2 · Definition
Absolute pressure is pressure measured relative to a perfect vacuum.
It includes the atmospheric contribution that gauge pressure omits and is required in the ideal gas law.
For example, a tire’s absolute pressure is its gauge reading plus the surrounding atmospheric pressure.
Absolute zero
Physics 2 · Definition
Absolute zero is zero kelvin, the lower limit of thermodynamic temperature in the models used here.
It corresponds to minus 273.15 degrees Celsius and does not mean all quantum motion disappears.
For example, an ideal-gas pressure-versus-Celsius graph can be extrapolated toward this limit, although a real gas may condense before reaching it.
Absorbance
Chemistry · Definition
Absorbance is a logarithmic measure of how much a sample reduces transmitted light at a specified wavelength.
It depends on the ratio of incident to transmitted intensity rather than simply being the fraction absorbed.
For example, absorbance 1 corresponds to transmission of one-tenth of the incident intensity.
Absorption spectrum
Physics 2 · Definition
An absorption spectrum shows radiation removed from a source spectrum as it passes through matter.
Atomic absorption lines correspond to photon energies that can promote atoms between allowed levels.
For example, a cooler gas in front of a continuous light source can produce dark lines at selected wavelengths.
Acceleration
Physics 1 · Physics 2 · Definition
Acceleration is the vector rate of change of velocity with time.
It can change an object’s speed, direction, or both, and it is measured in meters per second squared.
For example, a satellite moving at constant speed in a circular orbit accelerates toward the center.
Acceleration in calculus
Calculus AB · Calculus BC · Definition
Acceleration is the derivative of velocity with respect to time.
In one-dimensional motion, it is also the second derivative of position.
For example, s(t)=t² meters gives constant acceleration 2 meters per second squared.
A negative acceleration can increase speed when the velocity is also negative.
Accumulation function
Calculus AB · Calculus BC · Definition
An accumulation function uses a variable endpoint in a definite integral.
For example, A(x) defined as the integral of 2t from 0 to x equals x².
Where the integrand is continuous, the Fundamental Theorem of Calculus gives A′(x) equal to the integrand at x.
Variable endpoints that are themselves functions also require the chain rule.
Acid dissociation constant
Chemistry · Definition
An acid dissociation constant describes the equilibrium for proton donation by an acid in the specified solvent.
A larger value indicates greater proton-donation extent under comparable conditions.
For example, two weak acids at the same analytical concentration can produce different hydronium concentrations because their constants differ.
Acid–base indicator
Chemistry · Definition
An acid–base indicator has differently colored acid and conjugate-base forms.
Its color change occurs over a pH range, so selection must fit the titration's equivalence region.
For example, an indicator changing in a basic range can be suitable for some weak-acid–strong-base titrations.
Activation energy
Chemistry · Definition
Activation energy is the energy barrier associated with a reaction pathway in an introductory energy-profile model.
It concerns the pathway and is distinct from the reactant-to-product energy change.
For example, a thermodynamically favored reaction can remain slow when its activation barrier is large.
Adiabatic process
Physics 2 · Definition
An adiabatic process transfers no energy as heat across the chosen system boundary.
Work can still change internal energy, so adiabatic does not generally mean constant temperature.
For example, rapid compression of an insulated gas can warm it through positive work done on the gas.
Alcohol functional group
Chemistry · Definition
An alcohol contains a hydroxyl group attached to a saturated carbon atom.
This group can participate in hydrogen bonding, while the surrounding carbon structure also affects properties.
For example, ethanol is an alcohol that mixes readily with water.
Aldehyde
Chemistry · Definition
An aldehyde contains a carbonyl carbon bonded to at least one hydrogen.
This distinguishes it from a ketone, whose carbonyl carbon has two carbon-containing attachments.
For example, ethanal has a terminal carbonyl group with a hydrogen attached to its carbonyl carbon.
Aliphatic compound
Chemistry · Definition
Aliphatic compounds are organic compounds classified as nonaromatic.
They can contain open chains or nonaromatic rings and can be saturated or unsaturated.
For example, hexane and cyclohexane are both aliphatic hydrocarbons.
Aliquot
Chemistry · Definition
An aliquot is a measured portion taken from a sample.
In analysis, a representative aliquot lets a measured result be related to the original sample's total amount.
For example, a 10-milliliter portion of a uniform solution can be an aliquot of a larger prepared sample.
Alkane
Chemistry · Definition
An alkane is an acyclic saturated hydrocarbon with carbon–carbon single bonds.
Related saturated ring compounds are called cycloalkanes.
For example, propane is an alkane containing three carbon atoms.
Alkene
Chemistry · Definition
An alkene is a hydrocarbon containing a carbon–carbon double bond.
The double bond affects geometry and can create distinct arrangements that a formula alone does not specify.
For example, ethene is the simplest alkene.
Alkyl group
Chemistry · Definition
An alkyl group is a carbon–hydrogen substituent corresponding to removal of a hydrogen from an alkane.
It can form a branch on a larger organic structure.
For example, a methyl group is a one-carbon alkyl group attached through its carbon.
Alkyne
Chemistry · Definition
An alkyne is a hydrocarbon containing a carbon–carbon triple bond.
In a simple localized model, that triple bond contains one sigma bond and two pi bonds.
For example, ethyne has two carbons joined by a triple bond.
Alloy
Chemistry · Definition
An alloy is a metallic material containing more than one element.
Added atoms can replace host atoms or occupy spaces between them, changing properties such as hardness.
For example, carbon atoms in steel affect how easily iron atoms can move relative to one another.
Alpha decay
Physics 2 · Definition
Alpha decay emits a helium-4 nucleus containing two protons and two neutrons.
The daughter nucleus has an atomic number smaller by two and a mass number smaller by four.
For example, the emitted alpha particle is positively charged because it is a bare nucleus rather than a neutral helium atom.
Alternating series
Calculus BC · Definition
An alternating series has successive terms that switch between positive and negative signs.
The alternating series test establishes convergence when the term magnitudes eventually decrease and approach 0.
For example, 1−1/2+1/3−1/4+⋯ satisfies those conditions.
Alternating signs alone do not guarantee convergence.
Alternating series error bound
Calculus BC · Definition
For an alternating series satisfying the decreasing-to-zero magnitude conditions, the error after n terms is at most the next term’s magnitude.
The actual sum lies between two consecutive partial sums.
For example, stopping 1−1/2+1/3−⋯ after its tenth term gives error at most 1/11.
The bound requires the alternating series hypotheses and does not apply automatically to every signed series.
Alternating series test
Calculus BC · Definition
The alternating series test applies to alternating terms whose positive magnitudes eventually decrease to 0.
Those two conditions together ensure the partial sums approach a finite limit.
For example, 1−1/2+1/3−1/4+⋯ converges because 1/n decreases to 0.
If one condition fails, this test cannot by itself determine the result.
Alternative hypothesis
Statistics · Definition
An alternative hypothesis specifies the population difference or direction for which a test seeks evidence.
It should be chosen from the research question before inspecting results.
For example, an alternative can state that the population mean travel time is greater than 15 minutes.
Amine
Chemistry · Definition
An amine is an organic derivative of ammonia with one or more hydrogen atoms replaced by carbon-containing groups.
A nitrogen lone pair can allow many amines to act as bases.
For example, methylamine can accept a proton to form its conjugate acid.
Ammeter
Physics 2 · Definition
An ammeter measures current through a branch.
It is connected in series with that branch and ideally has negligible resistance so it changes the current as little as possible.
For example, inserting an ammeter into the wire leading to a resistor measures the current through that resistor.
Amphoteric species
Chemistry · Definition
An amphoteric species can behave as either an acid or a base.
An amphiprotic species specifically can donate or accept a proton in Brønsted reactions.
For example, bicarbonate can accept a proton to form carbonic acid or donate one to form carbonate.
Amplitude
Calculus AB · Calculus BC · Definition
The amplitude of a sinusoidal function is its maximum distance from the midline.
For y=A sin(Bx+C)+D, the amplitude is |A|.
For example, y=−3 cos x+2 has amplitude 3 and midline y=2.
Amplitude measures vertical variation rather than the interval required for one cycle.
Amplitude
Physics 1 · Physics 2 · Definition
Amplitude is the maximum magnitude of a repeating disturbance measured from its equilibrium value.
It describes the size of the oscillation rather than its frequency or propagation speed.
For example, pulling a spring-mounted mass farther from equilibrium increases its oscillation amplitude when it is released from rest.
Analyte
Chemistry · Definition
An analyte is the substance being identified or measured in an analysis.
Its amount or concentration is the target of the measurement rather than necessarily the largest component of the sample.
For example, an unknown acid is the analyte in an acid–base titration.
Angle addition identity
Calculus AB · Calculus BC · Definition
Angle addition identities express a trigonometric function of a sum using the separate angles.
For example, sin(a+b)=sin a cos b+cos a sin b.
Using a=π/6 and b=π/3 gives sin(π/2)=1.
The addition and subtraction identities also help rewrite expressions for limits, derivatives, and integrals.
Angular acceleration
Physics 1 · Physics 2 · Definition
Angular acceleration is the rate of change of angular velocity.
For a fixed axis its sign follows the chosen positive rotation direction, and its units are radians per second squared.
For example, a fan speeding up has angular acceleration, whereas one turning steadily does not.
Angular displacement
Physics 1 · Physics 2 · Definition
Angular displacement is the change in an object’s rotation angle about a specified axis.
Its sign follows the chosen rotation convention, and it is commonly measured in radians.
For example, a wheel can return to the same orientation after one turn while having an angular displacement of two pi radians.
Angular frequency
Physics 1 · Physics 2 · Definition
Angular frequency measures how rapidly the phase of a sinusoidal oscillation advances.
It equals two pi times the ordinary frequency and is measured in radians per second.
For example, two oscillators at the same ordinary frequency also have the same angular frequency even if their amplitudes differ.
Angular impulse
Physics 1 · Physics 2 · Definition
Angular impulse is the change in angular momentum caused by torque acting over time.
It is the signed area under a torque-versus-time graph about a consistent axis.
For example, applying a braking torque to a wheel for a longer interval changes its angular momentum more.
Angular momentum
Physics 1 · Physics 2 · Definition
Angular momentum describes motion about a specified point or axis.
For rigid rotation about a fixed principal axis it can be written as rotational inertia times angular velocity; a translating particle can also have angular momentum about a point.
For example, a ball moving past a pole has angular momentum about the pole even though the ball’s path is straight.
Angular velocity
Physics 1 · Physics 2 · Definition
Angular velocity describes the rate and direction of rotation about an axis.
For a fixed axis, a signed angular velocity is measured in radians per second, while its magnitude is angular speed.
For example, points on one rigid turntable share the same angular speed even though outer points have greater linear speeds.
Anion
Chemistry · Definition
An anion is a negatively charged ion.
It has more electrons than needed to balance its positive nuclear charge.
For example, a chlorine atom gaining one electron forms a chloride anion.
Anode
Chemistry · Definition
The anode is the electrode where oxidation occurs.
This definition applies to both galvanic and electrolytic cells.
For example, a zinc electrode releasing zinc ions and electrons is an anode in the zinc–copper galvanic model.
Antiderivative
Calculus AB · Calculus BC · Definition
An antiderivative of a function is another function whose derivative equals the original function on an interval.
Two antiderivatives on the same interval differ by a constant.
For example, x² is an antiderivative of 2x, and x²+7 is another.
An antiderivative is a function, whereas a definite integral is an accumulated value.
Antiderivative power rule
Calculus AB · Calculus BC · Definition
The antiderivative power rule integrates xⁿ to xⁿ⁺¹/(n+1)+C when n≠−1 on an interval where the powers are defined.
The exponent increases by one, and the result is divided by that new exponent.
For example, the indefinite integral of x² is x³/3+C.
The exceptional exponent −1 instead gives ln|x|+C on intervals excluding zero.
Antinode
Physics 2 · Definition
A wave antinode is a location where the specified oscillating quantity has its largest oscillation amplitude in a standing-wave pattern.
Displacement antinodes and pressure antinodes describe different quantities, and their instantaneous values can still pass through equilibrium.
For example, the center of a string in its fundamental two-fixed-end mode is a displacement antinode.
Aqueous solution
Chemistry · Definition
An aqueous solution is a solution in which water is the solvent.
The dissolved substance may originate as a solid, liquid, or gas, so aqueous does not mean that a solid has melted.
For example, sodium chloride dissolved in water is aqueous even though the pure salt is solid.
Arc length
Calculus BC · Definition
Arc length measures the distance along a curve over a specified parameter interval.
For a smooth graph y=f(x), its length is the integral of √(1+[f′(x)]²) with respect to x.
For example, the graph y=x from x=0 to x=1 has length √2.
Parametric curves instead integrate the magnitude of their velocity, accounting for how the curve is traced.
Archimedes’ principle
Physics 1 · Physics 2 · Definition
Archimedes’ principle states that a submerged object’s buoyant-force magnitude equals the weight of the fluid it displaces.
Only the submerged volume contributes to displaced fluid for a partly floating object.
For example, a floating boat displaces enough water for the water’s weight to match the boat’s total weight.
Area in polar coordinates
Calculus BC · Definition
Polar area accumulates thin sectors over a specified angle interval.
For a single boundary r(θ), the basic contribution is one half of r(θ)² dθ.
For example, r=2 over 0≤θ≤π/2 encloses area π when that interval traces the quarter-circle region once.
Check intersections, radial order, and repeated tracing before choosing the bounds or subtracting two squared radii.
Arithmetic sequence
Calculus BC · Definition
An arithmetic sequence has a constant difference between consecutive terms.
With first term a₁ and common difference d, its nth term is a₁+(n−1)d.
For example, 3,7,11,15 has common difference 4.
An infinite arithmetic sequence with nonzero common difference does not approach a finite limit.
Aromaticity
Chemistry · Definition
Aromaticity describes a special stability associated with particular cyclic, conjugated electron arrangements.
Having a ring alone does not make a compound aromatic.
For example, benzene is aromatic while cyclohexane is not.
Arrhenius equation
Chemistry · Definition
The Arrhenius equation models how a rate constant depends on absolute temperature and an activation-energy parameter.
Within the applicable model, increasing temperature raises the rate constant for a positive activation energy.
For example, a larger activation-energy parameter can make a reaction's rate constant more sensitive to a temperature change.
Atomic number
Physics 2 · Definition
Atomic number is the number of protons in an atomic nucleus.
It identifies the element and does not change merely when electrons are gained or lost.
For example, every carbon isotope has the same atomic number even though their neutron counts can differ.
Atomic orbital
Chemistry · Definition
An atomic orbital is a quantum state used to describe an electron in an atom.
Its spatial description gives a probability distribution rather than a definite path around the nucleus.
For example, an s orbital has a spherical spatial probability distribution, and one orbital can accommodate at most two electrons with opposite spins.
Aufbau principle
Chemistry · Definition
The Aufbau principle describes constructing a ground-state electron configuration by occupying lower-energy orbitals before higher-energy ones.
Orbital energies and some observed configurations require more care than a simple fixed filling diagram suggests.
For example, helium fills its 1s orbital before lithium begins occupying 2s.
Average rate of change
Calculus AB · Calculus BC · Definition
The average rate of change is the change in output divided by the change in input over a nonzero interval.
For a function f, it is [f(b)−f(a)]/(b−a).
For example, a temperature rising from 10°C to 16°C in 3 hours has average rate 2°C per hour.
It describes the whole interval and need not equal the rate at either endpoint.
Average value of a function
Calculus AB · Calculus BC · Definition
The average value of an integrable function on [a,b], with a<b, is its integral divided by b−a.
It is the height of a constant function with the same signed accumulation over that interval.
For example, x² on [0,3] has average value 9/3=3.
This is an average of outputs, whereas an average rate of change compares endpoint outputs with input change.
Average velocity
Physics 1 · Physics 2 · Definition
Average velocity is the displacement over an interval divided by the elapsed time.
It is a vector and need not equal the average speed, which uses total distance.
For example, a runner who returns to the starting line has zero average velocity for the full lap.
Avogadro’s constant
Chemistry · Definition
Avogadro’s constant connects a number of specified entities to an amount in moles.
Its exact value is 6.02214076 × 10²³ per mole.
For example, multiplying two moles of helium atoms by this constant gives their number of atoms.
Avogadro’s constant
Physics 2 · Definition
Avogadro’s constant is the number of specified entities per mole, approximately 6.02 × 10²³ per mole.
It converts between an amount in moles and a count of particles.
For example, doubling the number of moles in a sample doubles its particle count when the entity being counted stays the same.
Back-to-back stemplot
Statistics · Definition
A back-to-back stemplot compares two data sets using shared stems.
Leaves for one group go on one side and leaves for the other group go on the opposite side.
For example, two classes’ test scores can be compared while retaining their individual values.
Bar graph
Statistics · Definition
A bar graph uses separate bars to compare category counts or proportions.
The bar lengths or heights encode the values, and category order need not be numerical.
For example, separate bars can compare the number of votes for three lunch options.
Beats
Physics 2 · Definition
Beats are periodic changes in resultant amplitude when waves of nearby frequencies interfere.
The beat frequency equals the magnitude of the difference between those frequencies in the simple two-tone model.
For example, two slightly detuned tuning forks produce repeating louder and quieter sound.
Beer–Lambert law
Chemistry · Definition
The Beer–Lambert law relates absorbance to concentration, light-path length, and molar absorptivity under suitable conditions.
Comparisons require a controlled wavelength, consistent path length, and an appropriate calibration.
For example, doubling concentration doubles absorbance in the law's linear range if the other factors remain fixed.
Bernoulli’s equation
Physics 1 · Physics 2 · Definition
Bernoulli’s equation relates pressure, fluid speed, and elevation along a streamline in steady ideal flow.
Its usual form assumes an incompressible, nonviscous fluid and no added or removed mechanical energy along the path.
For example, in a horizontal pipe narrowing under those assumptions, higher flow speed accompanies lower pressure.
Beta-minus decay
Physics 2 · Definition
Beta-minus decay transforms a neutron into a proton while emitting an electron and an electron antineutrino.
The nucleus’s atomic number increases by one while its mass number stays fixed.
For example, the emitted electron is created in the nuclear transformation rather than being an orbital electron knocked free.
Beta-plus decay
Physics 2 · Definition
Beta-plus decay transforms a proton into a neutron while emitting a positron and an electron neutrino.
The nucleus’s atomic number decreases by one while its mass number stays fixed, and the reaction must satisfy its energy requirement.
For example, the emitted positron has the same mass as an electron but the opposite electric charge.
Bimodal distribution
Statistics · Definition
A bimodal distribution has two main peaks.
Two peaks can suggest different groups, but the graph alone does not establish their cause.
For example, combining commute times from nearby and distant neighborhoods may produce two peaks.
Binomial coefficient
Statistics · Definition
A binomial coefficient counts the ways to select a fixed number of objects from a set when order does not matter.
In a binomial probability calculation, it counts the possible success positions among the trials.
For example, choosing two success positions among four trials can be done in six ways.
Binomial distribution
Statistics · Definition
A binomial distribution models the number of successes in a fixed number of independent trials.
Every trial has two classified outcomes and the same probability of success.
For example, counting red cards in five independent draws with replacement follows a binomial model.
Biot–Savart law
Physics 2 · Definition
The Biot–Savart law builds the magnetic field of a steady current by adding contributions from small current elements.
Each contribution depends on current, separation, and the orientation of the current element relative to the observation point.
For example, summing contributions from a very long straight wire gives circular field lines around the wire.
Blackbody
Physics 2 · Definition
A blackbody is an ideal object that absorbs all incident electromagnetic radiation.
Its thermal emission spectrum depends on temperature, and at equilibrium it can emit and absorb equal powers without changing temperature.
For example, heating a blackbody shifts its peak emission toward shorter wavelengths and increases its total emitted power.
Blinding
Statistics · Definition
Blinding keeps specified participants or study personnel unaware of assigned treatments.
The study should state who is blinded because labels such as single-blind and double-blind can be used differently.
For example, both participants and outcome assessors can be kept unaware of which pill was assigned.
Bohr model
Physics 2 · Definition
The Bohr model describes a hydrogen electron using selected allowed orbits and quantized energies.
It connects electric attraction with quantum restrictions but is a limited model rather than a literal general picture of all atoms.
For example, it can account for hydrogen’s discrete line spectrum but does not fully describe multielectron atomic structure.
Bond dipole
Chemistry · Definition
A bond dipole describes a bond's separation of positive and negative charge.
Its direction and magnitude depend on the distribution of electron density.
For example, an O–H bond has a dipole because its electron density is drawn toward oxygen.
Bond enthalpy
Chemistry · Definition
A bond enthalpy is the enthalpy required to break a specified bond in the gas phase.
Tables often give average values, so estimates for complex reactions are approximate.
For example, a reaction estimate adds the costs of bonds broken and subtracts the energy released by bonds formed.
Bond order
Chemistry · Definition
Bond order describes the degree of bonding between two atoms in a specified bonding model.
Localized single, double, and triple bonds have orders one, two, and three, while delocalized models can give fractional values.
For example, equivalent bonds in a resonance description can have bonding intermediate between a single and double bond.
Bounded sequence
Calculus BC · Definition
A sequence is bounded when all its terms lie between two finite fixed numbers.
Being bounded alone does not guarantee that its terms approach one limit.
For example, the sequence (−1)ⁿ is bounded between −1 and 1 but does not converge.
A bounded monotone sequence does converge to a finite limit.
Boxplot
Statistics · Definition
A boxplot displays the median and quartiles with a box and extending whiskers.
A modified boxplot uses whiskers ending at the most extreme observations within the outlier fences and plots flagged observations separately.
For example, parallel boxplots can compare the centers and spreads of two classes’ scores.
Brønsted acid and base
Chemistry · Definition
A Brønsted acid donates a proton, and a Brønsted base accepts a proton.
These roles describe a particular reaction, so some substances can act in either role.
For example, water accepts a proton from hydrogen chloride but donates one to ammonia.
Buffer
Chemistry · Definition
A buffer contains appreciable amounts of a weak acid and its conjugate base, or the corresponding weak-base pair.
Those partners consume small added amounts of strong base or acid, limiting the pH change within the buffer's capacity.
For example, an acetic acid–acetate mixture can resist modest additions of acid or base.
Buffer capacity
Chemistry · Definition
Buffer capacity describes how much acid or base a buffer can absorb while keeping its pH change within a specified limit.
It depends on the amounts of both buffer partners as well as their ratio.
For example, a more concentrated buffer can generally absorb more added reagent than a dilute buffer with the same initial pH.
Bulk modulus
Physics 2 · Definition
Bulk modulus measures resistance to compression through pressure change divided by fractional volume decrease.
A larger modulus means a smaller volume change for the same pressure increase within the model’s range.
For example, a liquid with a high bulk modulus can often be approximated as incompressible in modest-pressure flow problems.
Buoyancy
Physics 1 · Physics 2 · Definition
Buoyancy is the net upward force arising from pressure differences over an object in a fluid under gravity.
Its magnitude equals the weight of the displaced fluid in the usual hydrostatic model.
For example, a fully submerged object can rise, sink, or remain suspended depending on how buoyancy compares with its weight.
Calorimetry
Chemistry · Definition
Calorimetry determines heat transfer from measured thermal changes in a specified setup.
The calculation accounts for the measured material and any supplied calorimeter heat capacity using an energy balance.
For example, warming solution can reveal heat released by a reaction when the reaction and surroundings are correctly distinguished.
Calorimetry
Physics 2 · Definition
Calorimetry measures energy transfers by tracking temperature changes and any relevant phase changes.
The energy balance must include the named objects and account for the container or energy leakage when they matter.
For example, mixing a warm metal sample with cooler water can help estimate the metal’s specific heat.
Capacitance
Physics 2 · Definition
Capacitance is the plate-charge magnitude divided by the potential difference for an ideal capacitor.
It is measured in farads and depends on geometry and the material between the conductors.
For example, increasing the overlap area of ideal parallel plates increases their capacitance when separation and dielectric stay fixed.
Capacitor
Physics 2 · Definition
A capacitor stores energy by separating charge between conducting regions.
In the ideal two-plate model, opposite plate charges create a potential difference related to charge by the capacitance.
For example, disconnecting a charged capacitor from a battery leaves energy stored until a discharge path is provided.
Carbonyl group
Chemistry · Definition
A carbonyl group contains a carbon atom double-bonded to oxygen.
Its surrounding attachments distinguish classes such as aldehydes, ketones, and carboxylic acids.
For example, the carbonyl carbon in acetone is attached to two carbon-containing groups.
Carboxylic acid
Chemistry · Definition
A carboxylic acid contains a carboxyl group with a carbonyl and a hydroxyl attached to the same carbon.
Its acidic proton can be transferred to a base, forming a carboxylate conjugate base.
For example, acetic acid forms acetate after proton donation.
Carrying capacity
Calculus BC · Definition
Carrying capacity is the positive equilibrium level built into a logistic growth model.
In y′=ky(1−y/K) with k>0, it is K.
For example, K=1000 models a population approaching 1000 when it starts between 0 and 1000.
It is a modeling parameter whose interpretation depends on the situation and assumptions.
Catalyst
Chemistry · Definition
A catalyst participates in a reaction pathway and is regenerated overall.
It can increase the reaction rate by providing a different pathway without changing the reaction's equilibrium constant at fixed temperature.
For example, a catalyst can help a mixture reach equilibrium sooner without changing the final equilibrium composition.
Categorical variable
Statistics · Definition
A categorical variable records a group or label for each individual.
Numeric labels still count as categories when arithmetic on them has no meaningful interpretation.
For example, favorite school subject is categorical, even if each subject receives a numbered code.
Cathode
Chemistry · Definition
The cathode is the electrode where reduction occurs.
It may deposit metal or host another reduction without necessarily gaining mass.
For example, a copper electrode receiving electrons that reduce dissolved copper ions is a cathode.
Cation
Chemistry · Definition
A cation is a positively charged ion.
Its total positive nuclear charge exceeds its total electron charge.
For example, calcium commonly forms a cation with charge +2 by losing two electrons.
Causation
Statistics · Definition
Causation means that changing one condition produces a change in an outcome.
Association alone is insufficient because confounding and other explanations may account for the pattern.
For example, a well-conducted randomized fertilizer experiment can support a causal conclusion about growth in its study setting.
Cell potential
Chemistry · Definition
Cell potential is an electrical potential difference associated with a specified cell reaction and conditions.
A reduction-potential table supplies reference values for half-reactions, and voltage is not multiplied when equation coefficients are scaled.
For example, balancing twice as many transferred electrons changes reaction amount bookkeeping but does not double the cell voltage.
Census
Statistics · Definition
A census attempts to collect data from every individual in the defined population.
It avoids sampling variability from selecting only a subset, but can still have coverage, response, and measurement errors.
For example, asking every enrolled student about transportation is a census of that enrollment.
Center of mass
Physics 1 · Physics 2 · Definition
The center of mass is the position obtained by weighting each part’s position by its mass.
Its motion represents the translational motion of the whole system under the net external force.
For example, adding a heavy object to one end of a light bar shifts the center of mass toward that end.
Central limit theorem
Statistics · Definition
For independent identically distributed observations with finite variance, the central limit theorem explains why the standardized sample mean approaches a Normal distribution as sample size increases.
There is no universal sample size that makes every population sufficiently close to Normal; skewness and extreme tails affect the approximation.
For example, averaging many independent waiting times can produce a less skewed distribution of averages than the original waiting-time distribution.
Editorial qualification: The captured n ≥ 30 statement is not a universal guarantee. Under the CLT’s assumptions, the accuracy of a Normal approximation depends on the population and sample size.
Centripetal acceleration
Physics 1 · Physics 2 · Definition
Centripetal acceleration is the inward acceleration component that changes the direction of motion along a curved path.
For circular motion its magnitude is speed squared divided by the circle’s radius.
For example, a car moving at constant speed around a bend still has centripetal acceleration.
Centripetal force
Physics 1 · Physics 2 · Definition
Centripetal force names the inward net-force component required for motion along a circular path.
It is supplied by actual interactions such as tension, friction, gravity, or normal forces rather than being an additional type of force.
For example, gravity supplies the centripetal force on a satellite in a circular orbit.
Chain rule
Calculus AB · Calculus BC · Definition
The chain rule differentiates a composition by multiplying the outer derivative at the inner output by the inner derivative.
For differentiable functions, (f(g(x)))′=f′(g(x))g′(x).
For example, the derivative of sin(x²) is cos(x²)·2x.
Nested compositions require a factor for each successive layer.
Charge density
Physics 2 · Definition
Charge density specifies how much electric charge is distributed per unit length, area, or volume.
The appropriate unit and geometry must be chosen before converting between density and total charge.
For example, spreading the same charge uniformly over a larger surface lowers its surface charge density.
Charging by induction
Physics 2 · Definition
Charging by induction uses a nearby charged object to redistribute charge without direct contact with that object.
A grounding connection or separation step can leave the target with net charge while conserving charge for the complete system.
For example, grounding a metal sphere while a negative rod is nearby can let electrons leave before the ground is disconnected.
Chemical equilibrium
Chemistry · Definition
Chemical equilibrium is a dynamic state in which forward and reverse reaction rates are equal.
Macroscopic composition remains constant in a closed system under fixed conditions, but reactant and product concentrations need not be equal.
For example, a reversible reaction can reach equilibrium with much more reactant than product.
Chi-square component
Statistics · Definition
A chi-square component is one cell’s contribution to a categorical test’s total chi-square statistic.
Examining components can show which cells disagree most with the null expectations, without automatically establishing a new significant comparison.
For example, a particularly large component highlights a category worth further contextual investigation.
Chi-square distribution
Statistics · Definition
A chi-square distribution is a nonnegative reference distribution whose shape depends on degrees of freedom.
It is the distribution of a sum of squared independent standard Normal variables.
For example, categorical-count tests compare a suitably constructed chi-square statistic with this reference distribution.
Chi-square goodness-of-fit test
Statistics · Definition
A chi-square goodness-of-fit test compares one categorical variable’s observed counts with a specified distribution.
When the null proportions are fully specified rather than estimated from the data, the degrees of freedom are the number of categories minus 1.
For example, a test can assess whether observed colors fit a manufacturer’s stated color proportions.
Chi-square statistic
Statistics · Definition
For categorical-count tests, the chi-square statistic adds each cell’s squared observed-minus-expected difference divided by its expected count.
Large contributions identify cells with substantial discrepancies relative to their expected counts.
For example, a cell observed 15 times when 10 are expected contributes 2.5 to the statistic.
Chi-square test for homogeneity
Statistics · Definition
A chi-square test for homogeneity compares a categorical variable’s distribution across independently sampled populations or randomized treatment groups.
The null model states that the category proportions are the same across groups.
For example, independently sampled students at several schools can be compared for their transportation-mode distributions.
Chi-square test of association
Statistics · Definition
A chi-square test of association examines the relationship between two categorical variables measured on one sample.
The null model states that the variables are independent in the population.
For example, one random sample of students can be classified by grade level and transportation mode to investigate association.
Chi-square testing
Statistics · Definition
Chi-square categorical tests compare observed counts with expected counts under a null model.
The appropriate test depends on the study design, categorical variables, and hypothesis, with independence and expected-count conditions checked.
For example, one sample with two categorical variables can be analyzed for association using a two-way table.
Chirality
Chemistry · Definition
Chirality is the property of an object that cannot be superimposed on its mirror image.
Molecules related as nonsuperimposable mirror images are enantiomers and can interact differently with other chiral species.
For example, a tetrahedral carbon bonded to four different groups can produce a pair of enantiomers.
Chromatography
Chemistry · Definition
Chromatography separates components through their differing interactions with mobile and stationary phases.
The measured movement or retention depends on the chosen materials and conditions.
For example, two dissolved dyes can travel different distances on a chromatographic strip.
Circuit node
Physics 2 · Definition
A circuit node is a connected conducting region treated as having one potential in the ideal-wire model.
Branches meet at nodes, so topology rather than the drawing’s shape determines how elements are connected.
For example, moving a wire on a schematic without changing its connections leaves the circuit nodes unchanged.
Cluster sampling
Statistics · Definition
Cluster sampling randomly selects groups of individuals rather than selecting individuals directly across the entire population.
In a one-stage cluster sample, all individuals in the selected clusters are included.
For example, researchers may randomly select classrooms and survey every student in those classrooms.
Coefficient of determination
Statistics · Definition
The coefficient of determination is the fraction of observed response variation accounted for by a fitted regression model.
In simple linear regression with an intercept, it equals the square of the correlation coefficient.
For example, a value of 0.64 means that the fitted linear model accounts for 64% of the response variation in those data.
Coefficient of friction
Physics 1 · Physics 2 · Definition
A coefficient of friction is a dimensionless parameter describing friction between a pair of surfaces in a specified model.
The static coefficient sets a maximum friction magnitude, whereas the kinetic coefficient models friction during sliding.
For example, changing a shoe’s material or the floor condition can change the coefficient without changing the shoe’s mass.
Coherence
Physics 2 · Definition
Coherence describes a sufficiently stable phase relationship between waves for a sustained interference pattern.
It depends on the sources and on the time or distance over which their phase relationship remains predictable.
For example, illuminating two slits with one suitable monochromatic source produces a stable relative phase at the slits.
Colligative property
Chemistry · Definition
Colligative properties depend chiefly on the number of dispersed solute particles in an appropriate dilute-solution model.
Familiar examples include boiling-point elevation and freezing-point depression.
For example, an ideally dissociated salt can affect freezing temperature more than the same molal amount of a nondissociating solute, though real concentrated solutions require additional care.
Collision model
Chemistry · Definition
The collision model explains reaction in terms of encounters between particles.
Effective encounters require sufficient energy and an appropriate arrangement, so not every collision produces reaction.
For example, increasing temperature can increase the fraction of collisions able to cross an energy barrier.
Combining random variables
Statistics · Definition
Combining random variables forms a new variable from values such as their sum or difference.
Means add or subtract, while the variance of a sum or difference is the sum of the variances when the variables are independent and have finite variances.
For example, two independent daily production counts have a total whose variance is the sum of their individual variances.
Common ratio
Calculus BC · Definition
A common ratio is the fixed multiplier between consecutive terms of a geometric sequence or series.
When the earlier term is nonzero, it equals the next term divided by that term.
For example, 2,6,18,54 has common ratio 3.
For a nonzero geometric series, its magnitude must be less than 1 for the series to converge.
Common-ion effect
Chemistry · Definition
The common-ion effect is an equilibrium response caused by adding a species already present in the relevant equilibrium.
In the simple dissolution model, adding a shared ion lowers the amount of solid that can dissolve at fixed temperature.
For example, added chloride can reduce silver chloride's solubility when other reactions are negligible.
Comparison tests
Calculus BC · Definition
Comparison tests determine convergence by relating nonnegative terms to a known benchmark series.
Direct comparison uses a correctly directed inequality, while limit comparison uses a positive finite limit of the term ratio.
For example, 0≤1/(n²+1)≤1/n² shows convergence because the upper p-series converges.
Being smaller than a divergent series or larger than a convergent one does not settle convergence.
Complement of an event
Statistics · Definition
The complement of an event contains every sample-space outcome outside that event.
Its probability is 1 minus the event’s probability.
For example, if a bus is on time with probability 0.8, the complementary event has probability 0.2.
Completely randomized design
Statistics · Definition
A completely randomized design assigns all experimental units to treatments using chance without first separating them into blocks.
The design uses comparison and replication to assess treatment differences.
For example, all eligible plants can be randomly allocated among three fertilizers.
Completing the square
Calculus AB · Calculus BC · Definition
Completing the square rewrites a quadratic using a squared linear expression plus a constant.
For a monic quadratic x²+bx+c, add and subtract (b/2)² to obtain (x+b/2)²+c−(b/2)².
For example, x²+4x+7 becomes (x+2)²+3.
The rewritten form can reveal a vertex or prepare an integral for substitution.
Complex ion
Chemistry · Definition
A complex ion is a charged coordination species containing a central atom or ion and bound ligands.
Ligand binding can change properties such as color and the amounts of free metal ions in solution.
For example, ammonia ligands can bind to a copper ion to form a differently colored complex.
Composite function
Calculus AB · Calculus BC · Definition
A composite function applies one function to the output of another.
In f(g(x)), evaluate g first and then use its output as the input to f.
For example, if f(u)=u² and g(x)=x+1, then f(g(x))=(x+1)².
The composition is defined only where the inner output lies in the outer function’s domain.
Compton scattering
Physics 2 · Definition
Compton scattering is the interaction in which a photon changes direction and transfers energy and momentum to an electron.
In the initially stationary free-electron model, a nonzero scattering angle gives the outgoing photon a longer wavelength.
For example, backward scattering gives a larger wavelength increase than forward scattering under the same assumptions.
Concavity
Calculus AB · Calculus BC · Definition
Concavity describes how a graph’s slope changes as the input increases.
A graph is concave up where its derivative increases and concave down where its derivative decreases.
For example, x² is concave up because its derivative 2x increases.
A positive second derivative establishes concave-up behavior on an interval, while a negative one establishes concave-down behavior.
Concentration cell
Chemistry · Definition
A concentration cell uses the same redox couple under differing activities in its two half-cells.
The concentration difference can produce a potential even though the standard half-cell potentials cancel.
For example, an ideal metal-ion concentration cell can operate in the direction that reduces the ion-concentration imbalance.
Condensation
Chemistry · Definition
Condensation is a phase change from gas to liquid.
It releases energy to the surroundings as particles form more closely associated liquid arrangements.
For example, water vapor condensing on a cooler surface forms liquid droplets.
Conditional convergence
Calculus BC · Definition
A series converges conditionally when it converges but the series of its absolute term values diverges.
Cancellation between positive and negative terms is essential to this convergence.
For example, 1−1/2+1/3−1/4+⋯ converges while the harmonic series of its magnitudes diverges.
Rearranging a conditionally convergent series can change its sum or destroy convergence.
Conditional probability
Statistics · Definition
Conditional probability measures the likelihood of one event when another event is known to have occurred.
It restricts attention to the conditioning event, which must have positive probability.
For example, knowing a drawn card is a face card changes the denominator when finding the probability it is a queen.
Conditional relative frequency
Statistics · Definition
A conditional relative frequency divides a cell count by the total for a specified row or column.
The denominator includes only individuals in the group being conditioned on.
For example, 6 walkers among 10 grade-11 students means that 60% of grade-11 students walk.
Confidence interval
Statistics · Definition
A confidence interval is a data-based range produced by a method with a specified long-run coverage rate for a population parameter.
For a fixed population, the calculated interval either contains the parameter or it does not; the confidence level describes the method over repeated samples.
For example, a 95% method captures the true parameter in about 95% of repeated intervals when its assumptions hold.
Confidence level
Statistics · Definition
A confidence level is the long-run fraction of intervals from a procedure that capture the true parameter under its assumptions.
Holding the data and procedure fixed, increasing the confidence level generally widens the interval.
For example, a 99% interval is usually wider than a 90% interval calculated from the same sample.
Confounding
Statistics · Definition
Confounding occurs when the effects of different variables cannot be distinguished in the observed comparison.
A potential explanation is that another variable is related to both the treatment or explanatory variable and the response.
For example, comparing fertilizer A outdoors with fertilizer B indoors mixes fertilizer and growing-location effects.
Conjugate acid–base pair
Chemistry · Definition
A conjugate acid–base pair consists of species differing by one proton.
Losing a proton forms an acid's conjugate base, while gaining one forms a base's conjugate acid.
For example, ammonium and ammonia form a conjugate acid–base pair.
Conservation of angular momentum
Physics 1 · Physics 2 · Definition
Angular momentum is conserved about a chosen axis when the net external angular impulse about that axis is zero.
Internal interactions can change the distribution of mass and the rotation rates of parts without changing the total.
For example, a rotating skater who pulls their arms inward can spin faster as their rotational inertia decreases.
Conservation of energy
Physics 1 · Physics 2 · Definition
Conservation of energy means total energy is unchanged when the full isolated system is considered.
Energy can transfer between objects or change form, so one selected energy store need not remain constant.
For example, friction can reduce a sliding block’s kinetic energy while increasing the thermal energy of the block and floor.
Conservative force
Physics 1 · Physics 2 · Definition
A conservative force does work that depends only on the starting and ending configurations.
A potential-energy function can describe it, and its work around a closed path is zero.
For example, near-Earth gravity does the same work between two heights whether a ball follows a vertical or a curved path.
Constant of integration
Calculus AB · Calculus BC · Definition
A constant of integration represents the arbitrary constant added when finding a family of antiderivatives.
It appears because differentiating a constant gives zero.
For example, every function x²+C has derivative 2x.
An initial condition may determine a particular value of C.
Constant-volume process
Physics 2 · Definition
A constant-volume process keeps a system’s volume fixed.
There is no pressure-volume work through the fixed boundary, although heat transfer can change internal energy.
For example, heating gas inside a sealed rigid vessel raises its temperature and pressure without moving its walls.
Constraint in optimization
Calculus AB · Calculus BC · Definition
A constraint restricts the quantities or choices permitted in a problem.
It can be an equation, an inequality, or a requirement from the physical context.
For example, a rectangle with perimeter 20 must satisfy 2l+2w=20 with positive side lengths.
Use constraints to determine the objective’s allowed domain before searching for extrema.
Constructive interference
Physics 2 · Definition
Constructive interference occurs when overlapping wave contributions reinforce one another.
For equal-frequency sinusoidal waves, in-phase contributions produce increased amplitude at the location considered.
For example, coherent light arriving with an integer-wavelength path difference can form a bright interference fringe.
Continuity
Calculus AB · Calculus BC · Definition
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree.
At a domain endpoint, continuity uses the relevant one-sided approach.
For example, x² is continuous at 3 because both its value and its nearby limit are 9.
Continuity on an interval requires the appropriate condition at every input in that interval.
Continuity equation
Physics 1 · Physics 2 · Definition
The continuity equation expresses conservation of mass in fluid flow.
For steady incompressible flow through one unbranched channel, cross-sectional area times average flow speed is constant.
For example, water moves faster through a narrower part of a pipe when the same volume passes each second.
Control group
Statistics · Definition
A control group supplies a comparison for evaluating an experimental treatment.
It may receive a placebo, no active treatment, or an established treatment, depending on the study’s design.
For example, a fertilizer experiment may compare a new fertilizer with the usual fertilizer rather than with no fertilizer.
Convection
Physics 2 · Definition
Convection transfers energy as a fluid moves from place to place.
The circulation can arise from density differences or be driven by a pump or fan.
For example, warmer water can rise while cooler water sinks, carrying thermal energy through a container.
Convenience sample
Statistics · Definition
A convenience sample includes individuals who are easiest to reach.
Easy access does not ensure that the sample represents the target population.
For example, surveying only students sitting nearest the classroom door is convenience sampling.
Conventional current
Physics 2 · Definition
Conventional current points in the direction that positive charge would move through a circuit.
In a metal wire, electrons usually drift in the opposite direction.
For example, a current arrow pointing right in a metal wire does not mean the conduction electrons drift right.
Convergence of a sequence
Calculus BC · Definition
An infinite sequence converges when its terms approach one finite value as the index tends to infinity.
It diverges when no finite limit exists.
For example, 1/n converges to 0, while (−1)ⁿ diverges by oscillating between two values.
A sequence’s convergence concerns its individual terms; a series uses the limits of their partial sums instead.
Convergence of a series
Calculus BC · Definition
A series converges when the sequence of its partial sums approaches a finite limit.
It diverges when there is no such finite limit.
For example, 1/2+1/4+1/8+⋯ converges to 1, while the harmonic series diverges.
Convergence of the individual terms to 0 is necessary but does not by itself establish convergence of their sum.
Core electron
Chemistry · Definition
Core electrons occupy inner shells and are usually less directly involved in bonding than valence electrons.
They help shield outer electrons from the nucleus's full positive charge.
For example, sodium's ten inner electrons form its neon-like core beneath its outer electron.
Correlation coefficient
Statistics · Definition
Pearson’s correlation coefficient measures the direction and strength of a linear relationship between two quantitative variables.
It ranges from −1 to 1, and a value near 0 does not rule out a strong curved relationship.
For example, points closely following a decreasing straight line have correlation near −1.
Corrosion
Chemistry · Definition
Corrosion is deterioration of a material through chemical or electrochemical reactions with its environment.
Metal corrosion commonly involves oxidation coupled to a reduction process.
For example, rust formation involves iron oxidation together with reactions involving oxygen and water.
Coulomb’s law
Physics 2 · Definition
Coulomb’s law gives the electric force between point charges using their charges and separation.
The magnitude grows with the product of the charge magnitudes and decreases with separation squared, with the medium accounted for in the stated model.
For example, doubling the separation of two fixed point charges reduces the force magnitude to one quarter.
Coupled reactions
Chemistry · Definition
Coupled reactions are linked through a mechanism or energy-transfer connection so that one can help drive another.
Adding favorable and unfavorable free-energy values is meaningful only when the processes are physically or chemically connected.
For example, a shared intermediate can connect an unfavorable transformation to a sufficiently favorable step.
Covalent bond
Chemistry · Definition
A covalent bond involves shared electron density between atoms.
The sharing can be approximately equal or unequal depending on the atoms and their environment.
For example, the bond in a hydrogen molecule shares electrons between two identical atoms.
Covalent network solid
Chemistry · Definition
A covalent network solid contains atoms joined by covalent bonds throughout an extended structure.
Changing or melting that structure requires more than separating intact small molecules.
For example, diamond contains a three-dimensional network of carbon–carbon bonds.
Critical angle
Physics 2 · Definition
The critical angle is the incidence angle at which the refracted ray would travel along the boundary toward a lower-index medium.
At larger incidence angles the ideal model gives total internal reflection.
For example, a ray leaving water for air can refract below the critical angle but undergo total internal reflection above it.
Critical point
Calculus AB · Calculus BC · Definition
A critical number is an input in the function’s domain where its derivative is zero or does not exist.
The corresponding point on the graph is called a critical point.
For example, x³ has a critical number at 0, but it has no local maximum or minimum there.
Critical numbers are candidates for extrema and need further analysis; inputs outside the domain are excluded.
Critical value
Statistics · Definition
A critical value is a cutoff from a reference distribution chosen for a confidence level or testing rule.
The relevant distribution, degrees of freedom, and one-sided or two-sided procedure determine the cutoff.
For example, a symmetric 95% Normal interval uses a cutoff near 1.96 standard-error units.
Cross product
Physics 1 · Physics 2 · Definition
A cross product combines two vectors to form a vector perpendicular to both.
Its magnitude depends on both magnitudes and the sine of their included angle, while reversing input order reverses its direction.
For example, a force whose line of action passes through the pivot gives zero torque cross product about that pivot.
Cross section
Calculus AB · Calculus BC · Definition
A cross section is the planar slice of a solid made by a specified cutting plane.
For volume by integration, its area must be expressed as a function of the position along the slicing direction.
For example, a square slice with side s(x) has area s(x)².
Integrating those areas over the allowed positions yields volume under the method’s assumptions.
Crystal lattice
Chemistry · Definition
A crystal lattice describes the repeating arrangement of structural sites in a crystalline solid.
The occupants can be ions, atoms, or molecules depending on the material.
For example, an ionic lattice places opposite charges near one another throughout a sodium chloride crystal.
Cumulative frequency
Statistics · Definition
Cumulative frequency adds counts through a specified value or interval.
It shows how many observations lie at or below that point in the ordered data.
For example, adding counts for all intervals up to 20 minutes gives the number of recorded waits no longer than 20 minutes.
Grouped cumulative tables follow their stated interval endpoints. The current Topic 1.7 lesson uses the empirical percentage at or below a data value, including equal observations. The preserved reading’s percentile formula instead counts strictly lesser observations. State the convention and keep the comparison consistent, especially with ties.
Cumulative relative frequency
Statistics · Definition
Cumulative relative frequency adds proportions through a specified value or interval.
A cumulative relative frequency graph shows how the accumulated proportion rises from 0 toward 1.
For example, a value of 0.7 at 20 minutes means that 70% of the recorded waits are at or below 20 minutes.
Grouped cumulative tables follow their stated interval endpoints. The current Topic 1.7 lesson uses the empirical percentage at or below a data value, including equal observations. The preserved reading’s percentile formula instead counts strictly lesser observations. State the convention and keep the comparison consistent, especially with ties.
Current efficiency
Chemistry · Definition
Current efficiency is the fraction of transferred charge that produces the specified electrode process.
Competing reactions or lost material can make the observed yield lower than the ideal charge-based prediction.
For example, an 80 percent efficiency means only 0.80 of the total charge contributes to the selected deposition process.
Cylindrical shell method
Calculus AB · Definition
The cylindrical shell method computes volume by accumulating thin cylindrical layers parallel to a rotation axis.
The volume contribution is circumference times shell height times thickness.
For example, rotating a vertical strip at x>0 around the vertical axis gives a contribution 2πx·height·dx.
Choose the radius as a distance from the axis and check the shell height against the bounding region.
Damping
Physics 2 · Definition
Damping is a process that transfers energy out of an oscillation and reduces its amplitude over time.
The motion can remain oscillatory or approach equilibrium without oscillating, depending on the damping strength.
For example, an oscillator moving through a viscous liquid loses mechanical energy to the liquid.
De Broglie wavelength
Physics 2 · Definition
De Broglie wavelength is the wavelength associated with a particle’s momentum in a quantum description.
It is Planck’s constant divided by momentum magnitude, so greater momentum corresponds to shorter wavelength.
For example, electron diffraction is noticeable when the experimental structure has a length scale comparable to the electrons’ de Broglie wavelength.
Decay constant
Physics 2 · Definition
A decay constant describes the decay probability per unit time in an exponential radioactive-decay model.
A larger constant corresponds to a shorter half-life for the same model.
For example, a fixed fractional loss per time interval produces a falling exponential parent-nucleus count rather than a constant absolute loss.
Definite integral
Calculus AB · Calculus BC · Definition
A definite integral accumulates a function over a specified interval as a limit of Riemann sums when that limit exists.
Its sign includes the function’s sign and the orientation of the limits.
For example, the integral of the constant −2 from 0 to 3 is −6, although the geometric area is 6.
In a rate model, the result has rate units multiplied by input units.
Degrees of freedom
Statistics · Definition
Degrees of freedom describe the independent information remaining after constraints or estimated quantities are accounted for.
The value depends on the procedure, rather than always being the sample size minus 1.
For example, a one-sample t procedure uses n − 1 degrees of freedom, while simple regression inference uses n − 2.
Deionized water
Chemistry · Definition
Deionized water has had most dissolved ions removed.
It is not necessarily free of every nonionic impurity or microorganism.
For example, using deionized water can reduce unintended ions in a prepared analytical solution.
Density curve
Statistics · Definition
A density curve models a continuous distribution with nonnegative height and total area 1.
Probability over an interval is the area under the curve across that interval, rather than its height at a single point.
For example, an interval containing one quarter of the total area has probability 0.25.
Derivative
Calculus AB · Calculus BC · Definition
A derivative gives the instantaneous rate of change of a function with respect to its input.
At a point, it is the finite limit of the difference quotient when that limit exists.
For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6.
A derivative function collects those rates across the inputs where the function is differentiable.
Destructive interference
Physics 2 · Definition
Destructive interference occurs when overlapping wave contributions oppose one another.
Complete cancellation requires equal-magnitude opposite disturbances at the relevant place and time.
For example, equal sinusoidal waves half a cycle out of phase can produce zero net displacement during their overlap.
Diamagnetism
Chemistry · Definition
Diamagnetism is a weak response opposing an applied magnetic field.
In the introductory classification, species with all electrons paired are described as diamagnetic when no stronger magnetic behavior dominates.
For example, neon has a filled electron configuration and is diamagnetic.
Diamagnetism
Physics 2 · Definition
Diamagnetism is an induced magnetic response opposing an applied magnetic field.
All materials have a diamagnetic contribution, but stronger magnetic responses can hide it in the net behavior.
For example, a material described as diamagnetic has a net response weakly opposed to the applied field.
Dielectric
Physics 2 · Definition
A dielectric is an insulating material that polarizes in an electric field.
In a capacitor it changes the field response and capacitance; charge or voltage changes depend on whether the battery remains connected.
For example, inserting a dielectric into an isolated charged capacitor can reduce its voltage while its free plate-charge magnitude remains fixed.
Dielectric constant
Physics 2 · Definition
A dielectric constant is a dimensionless measure of a material’s electric permittivity relative to vacuum under specified conditions.
In the ideal fully filled parallel-plate capacitor model, it gives the factor by which the material increases capacitance at fixed geometry.
For example, comparing otherwise identical capacitors with vacuum and a uniform dielectric gives a capacitance ratio equal to that dielectric constant.
Difference between sample means
Statistics · Definition
A difference between sample means compares two groups’ observed averages.
Inference must reflect whether the samples are independent or paired rather than treating every comparison the same way.
For example, two independently sampled classes’ mean travel times can estimate a difference between their population means.
Difference between sample proportions
Statistics · Definition
A difference between sample proportions compares the observed fractions in two samples.
Its order matters, and inference must account for both sample sizes and the independence or dependence of the groups.
For example, subtracting one class’s walking proportion from another’s estimates a difference between their population proportions.
Difference quotient
Calculus AB · Calculus BC · Definition
A difference quotient divides a change in a function’s output by the associated nonzero input change.
The form [f(a+h)−f(a)]/h is an average rate over an interval of length h.
For example, for f(x)=x² it simplifies to 2a+h when h≠0.
Taking its limit as h approaches 0 defines the derivative at a if the finite limit exists.
Differentiability
Calculus AB · Calculus BC · Definition
Differentiability means that a function has a finite derivative at the point under discussion.
Differentiability at a point implies continuity there, but continuity alone does not imply differentiability.
For example, |x| is continuous at 0 but its left and right tangent slopes differ, so it is not differentiable there.
Corners, cusps, vertical tangents, and discontinuities can prevent a finite derivative.
Differential
Calculus AB · Calculus BC · Definition
For a differentiable function y=f(x), the differential dy is defined as f′(x) dx.
It represents the tangent-line estimate of the output change associated with an input change dx.
For example, y=x² at x=3 and dx=0.01 gives dy=6·0.01=0.06.
The actual change in y can differ from this linear estimate.
Differential equation
Calculus AB · Calculus BC · Definition
A differential equation relates an unknown function to one or more of its derivatives.
A solution is a function that makes the equation true throughout an appropriate interval.
For example, y′=2x has solutions y=x²+C.
An initial condition can select a particular member of a family of solutions.
Diffraction
Physics 2 · Definition
Diffraction is the spreading and interference of waves near openings or obstacles.
The spreading is especially pronounced when the opening or obstacle size is comparable to wavelength.
For example, sound spreads through a doorway much more noticeably than visible light because sound wavelengths are far larger.
Diffraction grating
Physics 2 · Definition
A diffraction grating is an array of many regularly spaced openings or scattering regions.
Their coherent wave contributions produce narrow reinforced directions determined by spacing and wavelength.
For example, a grating spreads white light into colored orders because different wavelengths reinforce at different angles.
Diffusion
Chemistry · Definition
Diffusion is the net spreading of particles associated with random motion and concentration differences.
Individual particles move in many directions even as the overall distribution becomes more uniform.
For example, a gas initially concentrated in one part of a container spreads throughout the available space.
Dilution
Chemistry · Definition
Dilution lowers solute concentration by adding solvent while conserving the solute amount when no reaction or loss occurs.
Concentration calculations use the final solution volume rather than simply the volume of solvent added.
For example, increasing a solution's volume from 100 to 200 milliliters halves its concentration if its solute amount is unchanged.
Dipole–dipole forces
Chemistry · Definition
Dipole–dipole forces involve interactions between permanent molecular dipoles.
Favorable orientations place relatively positive and negative regions near one another.
For example, neighboring hydrogen chloride molecules can attract through their permanent dipoles.
Disc method
Calculus AB · Calculus BC · Definition
The disc method finds a solid’s volume using circular slices perpendicular to the rotation axis.
A slice with radius R has area πR².
For example, rotating the region under y=x on [0,1] around the horizontal axis gives volume equal to the integral of πx².
If the region leaves a hole around the axis, a washer rather than a full disc is needed.
Discontinuity
Calculus AB · Calculus BC · Definition
A discontinuity is a failure of continuity at an input being considered.
The obstruction can be a missing or mismatched value, unequal one-sided limits, oscillation, or unbounded behavior.
For example, assigning f(0)=5 while f(x)=x elsewhere creates a discontinuity at 0.
Identifying the obstruction determines whether changing one value can repair it.
Displacement
Physics 1 · Physics 2 · Definition
Displacement is the vector change in position from the start of an interval to its end.
It depends on those endpoints rather than the length or shape of the path traveled.
For example, walking around a closed track and returning to the start gives zero displacement but a nonzero distance traveled.
Distillation
Chemistry · Definition
Distillation separates mixture components using differences in their vaporization behavior.
Vapor is collected and condensed, with composition depending on the mixture and conditions.
For example, a vapor can be enriched in the more volatile component of a liquid mixture.
Distribution of data
Statistics · Definition
A distribution describes the values a variable takes and how often those values occur.
For quantitative data, useful features include shape, center, spread, and unusual observations.
For example, a distribution of commuting times shows which times are common and how widely they vary.
Domain
Calculus AB · Calculus BC · Definition
The domain of a function is the set of inputs for which the function is defined.
An expression and its context can both restrict the allowed inputs.
For example, the real function f(x)=1/(x−2) has domain all real numbers except 2.
For a model of elapsed time, a sensible domain may also require t≥0.
Doppler effect
Physics 2 · Definition
The Doppler effect is a difference between emitted and observed frequency caused by relative motion of source and observer.
For sound, both motions must be considered relative to the medium; motion changes the rate at which wavefronts are encountered.
For example, a sound source approaching a stationary listener usually produces a higher observed frequency than the source’s rest frequency.
Dotplot
Statistics · Definition
A dotplot places a dot for each observation above its numerical value.
Repeated values are usually shown by stacking dots.
For example, three observations equal to 4 produce a stack of three dots above 4.
Double angle identity
Calculus AB · Calculus BC · Definition
A double angle identity rewrites a trigonometric expression with twice an angle.
For example, cos(2θ)=2cos²θ−1 and sin(2θ)=2sin θ cos θ.
The cosine identity gives cos²θ=[1+cos(2θ)]/2, which can simplify a trigonometric integral.
These identities follow from the angle addition formulas.
Drag
Physics 1 · Physics 2 · Definition
Drag is a force from a fluid that opposes an object’s motion relative to that fluid.
Its dependence on speed, shape, and fluid properties varies with the physical regime.
For example, a parachute increases air drag by presenting a large area to the moving air.
Drift velocity
Physics 2 · Definition
Drift velocity is the small average carrier velocity associated with net charge flow in a conductor.
It is distinct from the carriers’ random microscopic motion and from the speed at which a circuit’s electromagnetic changes propagate.
For example, electrons can drift slowly along a wire while a circuit responds rapidly when a switch closes.
Effective nuclear charge
Chemistry · Definition
Effective nuclear charge describes the net nuclear attraction experienced by an electron after shielding is considered.
It depends on nuclear charge and the electron distribution rather than proton count alone.
For example, increasing effective attraction across a main-group period generally pulls valence electrons closer to the nucleus.
Effusion
Chemistry · Definition
Effusion is the escape of gas particles through an opening small enough that particle–particle collisions within the opening are negligible.
Under comparable conditions, lighter gases effuse faster on average than heavier gases.
For example, helium effuses faster than argon at the same temperature in the ideal model.
Elastic collision
Physics 1 · Physics 2 · Definition
An elastic collision conserves the total kinetic energy of the colliding system.
Total momentum is also conserved when the system has negligible net external impulse, although each object’s momentum and energy can change.
For example, ideal gas atoms redistribute their speeds in elastic collisions without changing the pair’s total kinetic energy.
Electric charge
Physics 2 · Definition
Electric charge is a conserved property that determines electric interactions.
It can be positive or negative and is measured in coulombs; ordinary net charges arise from an imbalance of charged particles.
For example, transferring electrons between initially neutral objects leaves one negatively charged and the other positively charged.
Electric current
Physics 2 · Definition
Electric current is the net rate of charge flow through a specified cross section.
It is measured in amperes, or coulombs per second, with direction defined by positive-charge motion.
For example, the same steady current passes through each element of a simple unbranched series loop.
Electric field
Physics 2 · Definition
An electric field assigns the electric force per unit positive test charge to each location.
The field is produced by the sources and is a vector, while the force on a negative charge points opposite the field.
For example, the electric field between ideal parallel capacitor plates can accelerate an electron opposite the field arrows.
Electric flux
Physics 2 · Definition
Electric flux measures the electric field’s normal contribution through a surface.
Its sign depends on the chosen surface normal, and its calculation includes field strength and orientation rather than a literal flow of matter.
For example, a uniform field parallel to a flat surface gives zero electric flux through that surface.
Electric force
Physics 2 · Definition
Electric force is the interaction of an electric field with a charged object.
For point charges at rest, like signs repel and opposite signs attract, with direction along their separation.
For example, an electron and proton exert equal-magnitude opposite forces on one another despite having different masses.
Electric polarization
Physics 2 · Definition
Electric polarization is a separation or rearrangement of positive and negative charge within otherwise neutral matter.
It can create induced dipoles without requiring a net charge to cross the object’s boundary.
For example, a nearby charged rod can slightly shift bound charges in an insulating object and produce attraction.
Electric potential
Physics 2 · Definition
Electric potential is electric potential energy per unit charge relative to a chosen reference.
It is a scalar measured in volts and can be positive, negative, or zero even where the electric field is nonzero.
For example, contributions from equal positive and negative source charges can cancel the potential at a point without canceling their field vectors.
Electric potential energy
Physics 2 · Definition
Electric potential energy is energy associated with the configuration of interacting charges.
Its value depends on the chosen reference, and a charge pair is counted once when calculating a system’s total pair energy.
For example, bringing two positive point charges closer requires positive external work in a slow controlled approach.
Electrical conductor
Physics 2 · Definition
An electrical conductor contains charge carriers that can move through the material.
In electrostatic equilibrium, mobile charges redistribute until the electric field within the conducting material is zero.
For example, excess charge on an isolated metal object resides on its surface after the charges settle.
Electrical insulator
Physics 2 · Definition
An electrical insulator strongly restricts the movement of charge carriers through its bulk.
Its bound charges can still rearrange slightly to polarize in an applied electric field.
For example, a plastic rod can retain localized excess charge even though it does not conduct current like a metal wire.
Electrical resistance
Physics 2 · Definition
Electrical resistance relates the potential difference across an element to the current through it.
It is measured in ohms and depends on the element’s material, geometry, and conditions.
For example, a longer uniform wire has greater resistance when cross-sectional area and material remain fixed.
Electrochemical cell
Chemistry · Definition
An electrochemical cell separates oxidation and reduction into electrode processes.
A galvanic cell delivers electrical energy from a favored reaction, while an electrolytic cell uses supplied electrical energy to drive an unfavored direction.
For example, a zinc–copper galvanic model transfers electrons through the external wire while ions carry charge through the solutions.
Electrolysis
Chemistry · Definition
Electrolysis uses an external electrical source to drive an electrode reaction in an otherwise unfavorable direction.
The products depend on the specified half-reactions and medium, including possible competition from water.
For example, reduction of metal ions at a cathode can produce an electroplated metal layer.
Electrolyte
Chemistry · Definition
An electrolyte forms mobile ions when dissolved and can allow a solution to conduct electric current.
Strong and weak electrolytes differ in the extent to which dissolved species form ions under the stated conditions.
For example, dissolved sodium chloride supplies sodium and chloride ions to water.
Electromagnetic induction
Physics 2 · Definition
Electromagnetic induction is the production of an emf associated with changing magnetic flux or with suitable motion through a magnetic field.
An induced current additionally requires a conducting closed path.
For example, moving a magnet toward a wire loop can induce an emf while holding it still produces none from constant flux.
Electromagnetic spectrum
Physics 2 · Definition
The electromagnetic spectrum is the continuous range of electromagnetic-wave frequencies and wavelengths.
Increasing frequency corresponds to decreasing vacuum wavelength, from radio toward gamma-ray wavelengths.
For example, infrared light has a longer wavelength and lower frequency than visible violet light.
Electromagnetic wave
Physics 2 · Definition
An electromagnetic wave contains oscillating electric and magnetic fields that transport energy.
In the plane-wave vacuum model, both fields are perpendicular to each other and to the propagation direction.
For example, visible light and radio waves can both travel through empty space.
Electromotive force
Physics 2 · Definition
Electromotive force is energy supplied per unit charge by a source or induction process.
Despite its name it is measured in volts, rather than newtons, and can exist even when no current flows.
For example, an ideal battery supplies an emf while an induced emf can be produced by changing magnetic flux.
Electron affinity
Chemistry · Definition
Electron affinity describes the energy change associated with adding an electron to a gaseous atom.
Tables may use different sign conventions, so the stated definition and sign must be checked.
For example, adding an electron to a chlorine atom releases energy under the usual energy-change convention.
Electron binding energy
Physics 2 · Definition
Electron binding energy is the energy required to remove an electron from its bound atomic state to the specified free-electron reference.
With a free electron at zero energy, a negative bound-state energy has a binding-energy magnitude equal to its distance below zero.
For example, an excited hydrogen atom requires less additional energy to ionize than the same atom in its ground state.
Electron configuration
Chemistry · Definition
An electron configuration describes how an atom's or ion's electrons occupy shells and subshells.
A ground-state configuration places electrons in the lowest-energy arrangement consistent with the electron-filling rules.
For example, neutral helium has two electrons in its 1s subshell.
Electron delocalization
Chemistry · Definition
Electron delocalization means that an electronic description extends across multiple atoms rather than being confined to one localized bond or atom.
It can affect bond equivalence and charge distribution.
For example, nitrate's bonding electrons are described across several nitrogen–oxygen positions.
Electron domain
Chemistry · Definition
An electron domain is a region of electron density around an atom in the VSEPR model.
A bond of any order counts as one domain, and a lone pair counts as another.
For example, two bonds and two lone pairs give water's oxygen four domains and a steric number of four.
Electron shielding
Chemistry · Definition
Electron shielding is the reduction in nuclear attraction experienced by an electron because other electrons also repel it.
Inner electrons generally shield an outer electron more effectively than electrons in the same shell.
For example, adding an occupied inner shell helps explain why atomic size usually increases down a periodic-table group.
Electron subshell
Chemistry · Definition
A subshell is a set of atomic orbitals with the same principal and angular-momentum quantum numbers.
The s, p, d, and f subshells contain one, three, five, and seven orbitals, respectively.
For example, a p subshell can accommodate six electrons because each of its three orbitals can hold two.
Electronegativity
Chemistry · Definition
Electronegativity describes an atom's tendency to attract shared electron density in a bond.
It helps predict bond polarity but is not the same as ionization energy or electron affinity.
For example, oxygen attracts shared electrons more strongly than hydrogen in an O–H bond.
Electronvolt
Physics 2 · Definition
An electronvolt is an energy unit equal to the elementary-charge magnitude multiplied by one volt.
It is approximately 1.60 × 10⁻¹⁹ joule and is not a unit of voltage.
For example, atomic transition energies are often more conveniently expressed in electronvolts than in very small fractions of a joule.
Electrostatic equilibrium
Physics 2 · Definition
Electrostatic equilibrium is a condition in which the charge distribution has settled and there is no sustained motion of free charges.
Within the material of an ideal conductor, the electric field is zero and the connected conductor is at one potential.
For example, charges on a metal surface rearrange until any tangential surface field that would drive them has vanished.
Elementary charge
Physics 2 · Definition
The elementary charge is the magnitude of the charge carried by a proton or electron, approximately 1.60 × 10⁻¹⁹ coulomb.
A proton carries positive elementary charge and an electron carries negative elementary charge.
For example, removing one electron from a neutral atom leaves it with a net charge of one positive elementary charge.
Elementary reaction
Chemistry · Definition
An elementary reaction is one step in a proposed mechanism rather than the sum of multiple steps.
For a specified elementary step, the rate-law concentration factors follow its reacting-particle event.
For example, a bimolecular elementary collision can have a rate proportional to the concentrations of its two reactants.
Emission spectrum
Physics 2 · Definition
An emission spectrum shows the frequencies or wavelengths of radiation emitted by a source.
An excited dilute gas produces characteristic bright lines associated with allowed energy-level transitions.
For example, comparing several observed lines with known element spectra can help identify which atoms are present.
Emissivity
Physics 2 · Definition
Emissivity describes a surface’s thermal-radiation emission compared with an ideal blackbody at the same temperature.
It is dimensionless and can depend on wavelength, surface condition, and temperature.
For example, two surfaces of equal area at the same temperature can radiate different powers because their emissivities differ.
Empirical formula
Chemistry · Definition
An empirical formula gives the simplest whole-number ratio of atoms in a compound.
It is obtained by comparing amounts in moles of each element rather than comparing their masses directly.
For example, both ethene and propene have the empirical formula CH₂ despite having different molecular formulas.
Endothermic process
Chemistry · Definition
An endothermic process absorbs heat into the chosen system.
Under constant-pressure conditions with only pressure–volume work, its enthalpy change is positive.
For example, melting ice absorbs heat while the phase change occurs.
Energy level
Physics 2 · Definition
An energy level is an allowed energy of a quantum system.
Bound systems have discrete levels, and a transition between them requires an energy transfer equal to the level difference.
For example, an atom emitting a photon drops to a lower energy level and transfers the difference to that photon.
Enthalpy
Chemistry · Definition
Enthalpy is a state function defined from internal energy, pressure, and volume.
At constant pressure with only pressure–volume work, an enthalpy change equals the heat transferred to the system.
For example, a negative reaction enthalpy indicates heat release for the written reaction under those conditions.
Entropy
Chemistry · Definition
Entropy is a state function related to the number of microscopic arrangements compatible with a macroscopic state.
Changes depend on both matter and energy distribution, so visual messiness alone is not a reliable measure.
For example, an ideal gas expanding at fixed temperature has more accessible positional arrangements and increased entropy.
Entropy
Physics 2 · Definition
Entropy is a thermodynamic state property related to how many microscopic arrangements are compatible with a macroscopic state.
The second law requires the total entropy of an isolated system not to decrease, although a subsystem’s entropy can decrease.
For example, energy spreading from a hot region into a cooler region of an insulated object increases the total entropy.
Equal variance
Statistics · Definition
An equal-variance condition means that the response spread around a regression line is the same across explanatory values.
A residual plot with a funnel-shaped pattern can challenge this assumption.
For example, wider residual scatter for older plants suggests that response variance increases with age.
Equilibrium constant
Chemistry · Definition
An equilibrium constant is the value of the reaction quotient at equilibrium for a specified reaction and temperature.
Its magnitude indicates equilibrium composition preference rather than reaction speed.
For example, a large constant often indicates a product-favored equilibrium, but the starting composition still determines the immediate net direction.
Equilibrium position
Physics 1 · Physics 2 · Definition
An equilibrium position is a position at which the net force in the modeled coordinate is zero.
In a stable oscillator, displacement to either side produces a restoring force back toward that position.
For example, a mass on a vertical spring oscillates about a position below the spring’s relaxed-length position because gravity shifts the balance.
Equipotential
Physics 2 · Definition
An equipotential is a set of locations with the same electric potential.
Moving a charge along it produces no electric potential-energy change, and electric field vectors are perpendicular to its surface.
For example, concentric spherical surfaces around an isolated point charge are equipotentials.
Equivalence point
Chemistry · Definition
The equivalence point is the condition where titrant and analyte have reacted in the balanced equation's stoichiometric ratio.
Its pH depends on the species remaining and is not always seven.
For example, a weak-acid–strong-base titration has a basic equivalence-point solution under the usual model.
Error bound
Calculus BC · Definition
An error bound gives a justified upper limit for the magnitude of an approximation’s error.
It states how far the approximate value can be from the true value.
For example, an error bound of 0.01 places the true value within 0.01 of the approximation.
A bound requires its stated hypotheses and is not a claim that the error actually equals the bound.
Escape speed
Physics 1 · Physics 2 · Definition
Escape speed is the minimum launch speed needed to reach indefinitely large separation in a specified gravitational model without further propulsion.
Its value depends on the launch position and attracting mass, with air resistance and other bodies omitted in the simplest model.
For example, launching from a greater distance from the same spherical planet requires a smaller escape speed.
Ester
Chemistry · Definition
A carboxylic ester has a carbonyl carbon bonded to an oxygen that also connects to a carbon-containing group.
It differs from a carboxylic acid because that linking oxygen does not carry the acid's proton.
For example, ethyl acetate is an ester containing an ethyl group attached through oxygen.
Ether
Chemistry · Definition
An ether has an oxygen atom bonded to two carbon-containing groups.
Its oxygen can accept hydrogen bonds, but an ether lacking an O–H group cannot donate one in the usual introductory model.
For example, dimethyl ether has an oxygen joining two methyl groups.
Euler’s method
Calculus BC · Definition
Euler’s method approximates a differential-equation solution by stepping along successive tangent lines.
For y′=F(x,y) with step size h, the update is yₙ₊₁=yₙ+hF(xₙ,yₙ).
For example, starting at (0,1) for y′=y with h=0.1 gives the next estimate y=1.1 at x=0.1.
Errors accumulate, and a smaller step often improves accuracy but does not make an approximation exact.
Evaporation
Chemistry · Definition
Evaporation is vaporization occurring at a liquid's surface.
It can occur below the boiling point as some particles escape into the gas phase.
For example, liquid water can evaporate at room temperature and leave less liquid behind.
Event in probability
Statistics · Definition
An event is a set of outcomes from a chance process’s sample space.
The event occurs when the observed outcome belongs to that set.
For example, rolling an even result is the event containing 2, 4, and 6 for a six-sided die.
Excited state
Physics 2 · Definition
An excited state is an allowed state with energy above the system’s ground state.
It can release energy through a transition to a lower state when an allowed process occurs.
For example, an excited atom can emit a photon whose energy equals the difference between two levels.
Exothermic process
Chemistry · Definition
An exothermic process transfers heat from the chosen system to its surroundings.
Under constant-pressure conditions with only pressure–volume work, its enthalpy change is negative.
For example, freezing water releases heat to the surroundings.
Expected count
Statistics · Definition
An expected count is the count predicted for a category or cell under the null model.
Goodness-of-fit expectations use the total count times the null proportion, while independence-table expectations use row total times column total divided by the grand total.
For example, a null proportion of 0.2 in 50 trials gives an expected count of 10.
Expected value
Statistics · Definition
Expected value is the probability-weighted mean of a random variable when that mean exists.
It describes a long-run average rather than a value that must occur on one trial.
For example, a fair die has expected value 3.5 even though no face is labeled 3.5.
Experiment
Statistics · Definition
An experiment deliberately imposes treatments and measures responses.
Random assignment, comparison, and replication help distinguish treatment effects from other variation.
For example, assigning plants to fertilizer treatments is an experiment.
Experimental factor
Statistics · Definition
A factor is an explanatory variable deliberately varied in an experiment.
Each assigned setting of that factor is called a level.
For example, fertilizer type is a factor with levels corresponding to the fertilizer choices.
Experimental unit
Statistics · Definition
An experimental unit is the smallest unit independently assigned a treatment.
A human experimental unit is often called a subject or participant.
For example, if fertilizer is assigned to an entire planter, the planter rather than each leaf is the experimental unit.
Explanatory variable
Statistics · Definition
An explanatory variable is used to predict or explain variation in a response variable.
In an experiment, researchers deliberately assign values or treatments related to an explanatory variable.
For example, assigned study time could be an explanatory variable when test score is the response.
Exponential growth and decay
Calculus AB · Calculus BC · Definition
An exponential growth or decay model makes a quantity’s instantaneous rate proportional to the quantity itself.
The differential equation y′=ky gives y=Ceᵏˣ, with positive k describing growth and negative k describing decay for positive y.
For example, a population with a constant relative growth rate doubles in equal time intervals.
Its relative rate stays constant, while its absolute rate changes with the amount present.
Exponential model
Statistics · Definition
An exponential model multiplies the response by the same factor for each equal increase in the explanatory variable.
For positive responses, plotting log y against x can produce a straight line.
For example, a population that doubles every year follows an exponential rather than a power model of time.
External force
Physics 1 · Physics 2 · Definition
An external force is exerted on part of a chosen system by something outside that system.
The sum of external forces determines the rate of change of the system’s total momentum.
For example, Earth’s gravitational force is external to a system consisting only of a falling ball.
Extrapolation
Statistics · Definition
Extrapolation uses a model to predict beyond the range of explanatory values used to fit it.
A pattern observed within the data range may not continue outside that range.
For example, extending a short-term plant-growth line far into the future may give unrealistic heights.
Extreme Value Theorem
Calculus AB · Calculus BC · Definition
The Extreme Value Theorem guarantees an absolute maximum and absolute minimum for a function continuous on a closed bounded interval.
The extrema can occur at endpoints or interior points.
For example, x² on [−1,2] attains minimum 0 at 0 and maximum 4 at 2.
An open interval or a missing continuity hypothesis does not carry the same guarantee.
Factor level
Statistics · Definition
A factor level is a particular setting of an experimental factor.
Combining the levels of different factors determines the treatment conditions.
For example, four hours and eight hours can be two levels of a light-duration factor.
Factorial
Calculus BC · Definition
The factorial n! multiplies the positive integers from 1 through n for a positive integer n.
By definition, 0!=1.
For example, 4!=4·3·2·1=24.
Factorials occur in Taylor coefficients and their rapid growth often makes the ratio test useful.
Faraday’s constant
Chemistry · Definition
Faraday’s constant is the magnitude of electric charge carried by one mole of electrons.
It is approximately 96485 coulombs per mole and connects measured charge to the amount of electrons transferred.
For example, depositing one mole of a metal from ions with charge +2 requires two moles of electrons in the ideal model.
Faraday’s law
Physics 2 · Definition
Faraday’s law relates induced emf to the rate of change of magnetic flux through a circuit.
Changing field strength, loop area, or loop orientation can change flux, while a large constant flux alone produces no emf.
For example, more identical coil turns produce a larger emf when each turn experiences the same flux-change rate.
Ferromagnetism
Physics 2 · Definition
Ferromagnetism is a strong collective magnetic response associated with aligned domains.
Some ferromagnetic materials can retain substantial magnetization after an applied field is removed.
For example, iron can become a permanent magnet when enough domain orientations remain aligned.
Fibonacci sequence
Calculus BC · Definition
A Fibonacci sequence begins with specified starting terms and obtains each later term by adding the preceding two.
A common starting convention uses 0 and 1.
For example, that convention gives 0,1,1,2,3,5,8, and so on.
Because the indexing convention can vary, check the stated first terms before identifying a particular indexed term.
Field line
Physics 2 · Definition
A field line is a drawn curve whose tangent shows the local direction of a vector field.
Closer spacing conventionally represents greater field strength, and lines are a representation rather than physical threads.
For example, electric field lines point away from a positive point charge and toward a negative one.
Filtration
Chemistry · Definition
Filtration separates suspended particles from a fluid using a barrier that retains appropriate particle sizes.
It generally does not remove individually dissolved ions or molecules.
For example, a filter can separate an undissolved precipitate from the surrounding solution.
First derivative test
Calculus AB · Calculus BC · Definition
The first derivative test applies to a function continuous at a critical input and differentiable on nearby intervals on both sides.
A positive-to-negative sign change gives a local maximum, and a negative-to-positive change gives a local minimum.
For example, the derivative 2x of x² changes from negative to positive at 0, identifying a local minimum.
If the sign does not change, this test does not identify a local extremum there.
First law of thermodynamics
Physics 2 · Definition
The first law of thermodynamics expresses conservation of energy for a thermodynamic system.
For the closed gas model used here, internal-energy change equals heat transferred into the gas plus work done on it.
For example, compressing an insulated gas can increase its internal energy even though no heat enters.
Five-number summary
Statistics · Definition
A five-number summary contains the minimum, first quartile, median, third quartile, and maximum.
These values summarize the location and spread of ordered numerical data.
For example, the summary 2, 4, 7, 10, and 15 identifies a median of 7 and an IQR of 6.
Fluid
Physics 1 · Physics 2 · Definition
A fluid is a material that flows and continuously deforms under a sustained shear stress.
Liquids and gases are both fluids, although their compressibility and other properties can differ greatly.
For example, both water in a pipe and air flowing through a duct are fluids.
Focal length
Physics 2 · Definition
Focal length is the signed distance from the relevant lens or mirror reference point to its principal focal point in the chosen convention.
It determines how strongly the optical element converges or diverges paraxial rays.
For example, a converging lens with a shorter positive focal length bends initially parallel rays to a nearer focus.
Focal point
Physics 2 · Definition
A focal point is where rays initially parallel to the principal axis converge or appear to diverge after reflection or refraction.
The principal-axis description uses the paraxial model, in which rays remain close to the axis.
For example, a converging thin lens brings parallel incoming rays to its far focal point.
Force
Physics 1 · Physics 2 · Definition
A force describes an interaction that can change an object’s momentum.
It is a vector measured in newtons, so both size and direction matter when forces are added.
For example, a rope can pull a cart while the floor exerts friction in another direction.
Formal charge
Chemistry · Definition
Formal charge is electron bookkeeping that assigns half of each bonding pair to each bonded atom.
It compares that assigned electron count with the neutral atom's valence-electron count and is not a direct measurement of actual charge distribution.
For example, all formal charges in an ion's Lewis structure must sum to the ion's total charge.
Formula unit
Chemistry · Definition
A formula unit specifies the simplest whole-number ratio of ions in an ionic compound.
It is a counting unit for an extended solid rather than an isolated molecule.
For example, one calcium chloride formula unit represents one calcium ion and two chloride ions.
Fourier series
Calculus BC · Definition
A Fourier series represents periodic behavior using constant, sine, and cosine components with related frequencies.
The coefficients measure those components through integrals over a period.
For example, a square wave can be approximated by adding suitable odd sine harmonics.
Convergence depends on the function; under common regularity conditions, the series at a jump approaches the average of the one-sided values.
Free fall
Physics 1 · Physics 2 · Definition
Free fall is motion in which gravity is the only significant force acting on the object.
An object can be in free fall while moving upward, downward, or sideways.
For example, a thrown ball is approximately in free fall after leaving the hand when air resistance is negligible.
Free-body diagram
Physics 1 · Physics 2 · Definition
A free-body diagram shows the external forces acting on one selected object or system.
Each arrow names a real interaction, and its direction is the direction of that force.
For example, a book resting on a desk has a downward gravitational force and an upward normal force; velocity is not an extra force arrow.
Frequency
Physics 1 · Physics 2 · Definition
Frequency is the number of complete repeating cycles per unit time.
It is measured in hertz, meaning cycles per second, and is the reciprocal of the period.
For example, a vibrating string with a higher frequency produces more cycles each second.
Frequency table
Statistics · Definition
A frequency table lists the number of observations in each category or numerical interval.
Its counts add to the total number of observations when the groups do not overlap and cover all observations.
For example, a table can record how many students chose each lunch option.
Function
Calculus AB · Calculus BC · Definition
A mathematical function assigns exactly one output to each input in its domain.
Inputs can share an output, but one input cannot have two different outputs.
For example, f(x)=x² assigns the output 9 to both 3 and −3.
Function notation names the rule and its input; f(3) means the output when the input is 3.
Functional group
Chemistry · Definition
A functional group is a characteristic atom arrangement associated with patterns of chemical behavior in organic compounds.
Its effect depends on the rest of the molecule as well as the group's identity.
For example, an alcohol's hydroxyl group can support hydrogen bonding.
Fundamental Theorem of Calculus
Calculus AB · Calculus BC · Definition
The Fundamental Theorem of Calculus connects accumulation with differentiation.
For a continuous integrand, differentiating its accumulation function returns the integrand.
It also allows a definite integral of a continuous function to be evaluated by the endpoint change of an antiderivative.
For example, the integral of 2x from 1 to 3 equals 3²−1²=8.
Gamma emission
Physics 2 · Definition
Gamma emission releases a high-energy photon when an excited nucleus moves to a lower nuclear energy state.
It changes neither the proton count nor the neutron count, so the isotope can remain the same.
For example, a daughter nucleus produced in another decay can subsequently emit a gamma photon as it loses excitation energy.
Gamma radiation
Physics 2 · Definition
Gamma radiation consists of high-energy electromagnetic photons, commonly produced in nuclear or particle processes.
The photons carry energy and momentum but have no electric charge or rest mass; the radiation’s name alone does not specify a decay mode.
For example, a gamma photon belongs to the same electromagnetic spectrum as a radio photon but has a much higher frequency and energy.
Gauge pressure
Physics 1 · Physics 2 · Definition
Gauge pressure is the difference between a system’s pressure and a specified surrounding reference pressure.
An ordinary pressure gauge usually uses local atmospheric pressure as its zero.
For example, a container open to the atmosphere has approximately zero gauge pressure while its absolute pressure is not zero.
Gauss’s law
Physics 2 · Definition
Gauss’s law relates net electric flux through a closed surface to the net charge enclosed by that surface.
The law always holds in its applicable electromagnetic model, while using it to calculate a field simply requires sufficient symmetry.
For example, a charge outside a closed surface contributes zero net electric flux through the whole surface even though its field can be nonzero on parts of it.
General addition rule
Statistics · Definition
The general addition rule finds the probability that at least one of two events occurs.
It adds their probabilities and subtracts the probability that both occur, avoiding double counting.
For example, the probability of a card being red or a king must subtract the red kings counted twice.
General multiplication rule
Statistics · Definition
The general multiplication rule finds the probability that two events both occur.
It multiplies the first event’s probability by the second event’s probability conditional on the first.
For example, two red draws without replacement use the updated red proportion for the second draw.
General solution
Calculus AB · Calculus BC · Definition
A general solution describes a family of functions satisfying a differential equation.
Arbitrary constants account for the freedom left by the equation.
For example, y=x²+C is a general solution of y′=2x.
Separation or other manipulations can miss special solutions, so the family and any excluded cases need checking.
Geometric distribution
Statistics · Definition
A geometric distribution models the trial number of the first success in independent trials with a constant success probability.
In this convention its possible values start at 1, and earlier trials must all be failures.
For example, the number of draws with replacement until the first red card has a geometric distribution.
Geometric sequence
Calculus BC · Definition
A geometric sequence multiplies each term by the same fixed ratio to obtain the next.
With first term a₁ and ratio r, its nth term is a₁rⁿ⁻¹.
For example, 8,4,2,1 has common ratio 1/2 and approaches 0.
This list of terms is distinct from the geometric series obtained by adding them.
Geometric series
Calculus BC · Definition
A geometric series adds terms that have a fixed common ratio.
With first term a and ratio r, it converges to a/(1−r) when |r|<1.
For example, 1+1/2+1/4+⋯ sums to 2.
For a nonzero first term and |r|≥1, the infinite series diverges.
Gibbs free energy
Chemistry · Definition
Gibbs free energy combines enthalpy and entropy contributions to a system at specified temperature and pressure.
For a reaction at constant temperature and pressure, a negative actual reaction free-energy change favors its forward direction, while standard and actual values must be distinguished.
For example, a favorable reaction can still be slow if its pathway has a large activation barrier.
Gibbs phenomenon
Calculus BC · Definition
The Gibbs phenomenon is persistent overshoot near a jump when a function is approximated by finite Fourier sums.
Adding terms narrows the region of the overshoot but does not make its peak relative to the jump disappear.
For example, Fourier approximations to a square wave can rise above its high level near a transition.
This local behavior does not mean that every Fourier series diverges everywhere.
Graph reflection
Calculus AB · Calculus BC · Definition
A graph reflection flips points across a specified line.
Replacing f(x) by −f(x) reflects across the horizontal axis, while f(−x) reflects across the vertical axis.
For example, −x² is the reflection of y=x² across the horizontal axis.
Reflecting across y=x interchanges input and output coordinates and can produce an inverse relation.
Graph stretching and compression
Calculus AB · Calculus BC · Definition
Graph scaling multiplies output values or changes how quickly the input is traversed.
For y=A f(x), a factor |A|>1 stretches vertically, while 0<|A|<1 compresses vertically.
For example, f(2x) compresses the graph horizontally by a factor of 1/2 relative to f(x).
Negative factors also introduce reflections, and a zero factor is a separate degenerate case.
Graph translation
Calculus AB · Calculus BC · Definition
A graph translation shifts every point by the same horizontal and vertical amounts.
The graph y=f(x−h)+k moves the original graph h units right and k units up.
For example, y=(x−2)²+3 shifts y=x² right 2 and up 3.
Negative shifts move in the opposite directions.
Gravitational field
Physics 1 · Physics 2 · Definition
A gravitational field gives the gravitational force per unit mass at a location.
Its direction is the direction of the force on a positive test mass, and its units can be written as newtons per kilogram.
For example, near Earth the field points approximately downward, whether or not a ball is placed there.
Gravitational potential energy
Physics 1 · Physics 2 · Definition
Gravitational potential energy is energy associated with the relative positions of attracting masses.
Near Earth, changes can be approximated by mass times gravitational field strength times height change when that field is nearly constant.
For example, lifting a book increases the gravitational potential energy of the book–Earth system.
Gravity
Physics 1 · Physics 2 · Definition
Gravity is the attractive interaction between masses in the Newtonian model.
For spherical bodies or point-mass approximations, its strength decreases with the square of the separation between their centers.
For example, Earth’s gravity provides the inward force that keeps a satellite in orbit.
Ground state
Physics 2 · Definition
The ground state is the lowest-energy state available to the specified quantum system.
Its energy need not be zero because the energy reference is chosen separately.
For example, a hydrogen atom in its ground state is more tightly bound than the same atom in an excited state.
Grounding
Physics 2 · Definition
Grounding connects a conductor to a large charge reservoir that can receive or supply charge.
The connection lets charge redistribute to satisfy the potential conditions; it does not always make an object neutral in the presence of nearby sources.
For example, a grounded sphere near a charged rod can acquire net charge from the reservoir.
Half angle identity
Calculus AB · Calculus BC · Definition
A half angle identity expresses a trigonometric function of half an angle using the original angle.
For example, sin²(θ/2)=(1−cos θ)/2.
Taking a square root gives a magnitude, and the sign of sin(θ/2) depends on the angle’s quadrant.
For θ=π, the formula yields sin(π/2)=1 using the positive sign.
Half-life
Calculus AB · Calculus BC · Definition
A half-life is the time required for a positive exponentially decaying quantity to fall to half its current amount.
For y(t)=y₀eᵏᵗ with k<0, it equals ln 2 divided by −k.
For example, a quantity with half-life 3 hours falls to one quarter of its initial amount after 6 hours.
The same half-life applies from any starting time within the exponential model.
Half-life
Physics 2 · Definition
Half-life is the time in which an ensemble of undecayed parent nuclei is expected to decrease to half its initial number.
It is a statistical property of the isotope and does not predict the decay time of one nucleus.
For example, after two half-lives a large ideal sample has about one quarter of its original parent nuclei remaining.
Half-reaction
Chemistry · Definition
A half-reaction describes either the oxidation or reduction part of a redox process.
It includes electrons explicitly so that charge and electron transfer can be balanced.
For example, zinc forming a zinc ion and two electrons is an oxidation half-reaction.
Hall effect
Physics 2 · Definition
The Hall effect is the transverse charge separation produced when moving carriers in a conductor experience a magnetic force.
The separation creates an electric field and voltage that can balance the sideways magnetic force.
For example, reversing the applied magnetic field reverses the side on which a given carrier sign accumulates.
Harmonic series
Calculus BC · Definition
The harmonic series is 1+1/2+1/3+1/4+⋯.
It diverges even though its individual terms approach 0.
For example, grouping terms from 1/2 through 1/3+1/4 and then 1/5+⋯+1/8 shows repeatedly added blocks of at least 1/2.
It is a standard comparison that separates a necessary term condition from actual convergence.
Head loss
Physics 2 · Definition
Head loss describes a decrease in a flowing fluid’s mechanical energy per unit weight.
Friction and fittings can produce losses that the ideal Bernoulli model omits.
For example, a long rough pipe can require a greater pressure difference than a short smooth one to maintain the same flow.
Heat
Physics 2 · Definition
Heat is energy transferred between systems because of a temperature difference.
It describes a transfer rather than a substance or energy store contained in an object.
For example, when a warmer mug transfers energy to a cooler hand, the mug loses internal energy and the hand gains it.
Heat capacity
Chemistry · Definition
Heat capacity is the heat required to raise a specified object's temperature by one temperature unit under stated conditions.
It refers to the entire object and differs from specific heat capacity, which is expressed per unit mass.
For example, doubling the amount of the same material approximately doubles its heat capacity.
Heat capacity
Physics 2 · Definition
Heat capacity is the energy needed to raise the temperature of an entire object by one temperature unit under specified conditions.
For a uniform sample in the simple constant-specific-heat model, it equals mass times specific heat.
For example, twice as much of the same material has twice the heat capacity when its state and specific heat are unchanged.
Heisenberg uncertainty principle
Chemistry · Definition
The Heisenberg uncertainty principle places an intrinsic limit on simultaneously specifying a particle's position and momentum.
It is a property of quantum states rather than merely a problem caused by poor measuring instruments.
For example, atomic electrons are described with probability distributions instead of exact classical trajectories around a nucleus.
Henderson–Hasselbalch equation
Chemistry · Definition
The Henderson–Hasselbalch equation relates a buffer's pH to an acid's pKa and the ratio of conjugate-base to acid activities.
Concentration ratios approximate activity ratios only when the model's conditions are appropriate.
For example, equal effective amounts of an acid and its conjugate base give pH approximately equal to pKa.
Henry’s law
Chemistry · Definition
Henry’s law relates a gas's dissolved concentration to its partial pressure over the solution at fixed temperature in an appropriate dilute model.
The proportionality constant depends on the gas, solvent, temperature, and the convention used to write the relation.
For example, increasing carbon dioxide's partial pressure can increase its dissolved concentration when the model applies.
Hess’s law
Chemistry · Definition
Hess’s law states that an overall enthalpy change is independent of the reaction path.
Balanced step equations can be added while their enthalpy changes are added with matching reversals and scaling.
For example, reversing a step changes the sign of its enthalpy before it is included in a sum.
Higher-order derivative
Calculus AB · Calculus BC · Definition
A higher-order derivative is obtained by differentiating a function repeatedly.
The nth derivative means that the derivative operation has been applied n times.
For example, f(x)=x³ has third derivative 6 and fourth derivative 0.
Each differentiation requires the preceding derivative to exist on the region being considered.
Histogram
Statistics · Definition
A histogram groups quantitative observations into numerical intervals and displays their frequencies or proportions.
Neighboring bars represent adjacent intervals, so the graph preserves numerical order.
For example, a histogram can group travel times into ten-minute intervals.
Histogram class interval
Statistics · Definition
A histogram class is a numerical interval used to group observations.
The class width determines the span of values within that interval and affects the displayed shape.
For example, intervals from 0 to 10 and from 10 to 20 minutes have width 10 minutes.
Hooke’s law
Physics 1 · Physics 2 · Definition
Hooke’s law models an ideal spring force as proportional to deformation and opposite in direction.
The deformation is measured from the spring’s relaxed length, and the law applies only within the spring’s approximately elastic linear range.
For example, doubling a small extension doubles the restoring-force magnitude of a spring that obeys this law.
Horizontal asymptote
Calculus AB · Calculus BC · Definition
A horizontal asymptote y=L describes a finite limiting output as the input tends to positive or negative infinity.
The two ends of a graph may have different horizontal asymptotes.
For example, 1/x has horizontal asymptote y=0 at both ends.
A graph may cross a horizontal asymptote at finite inputs.
Hund’s rule
Chemistry · Definition
Hund’s rule describes the lowest-energy arrangement within a set of degenerate orbitals.
Electrons occupy those orbitals singly with parallel spins before pairing.
For example, nitrogen's three 2p electrons occupy three different p orbitals in its ground state.
Hydrocarbon
Chemistry · Definition
A hydrocarbon is a compound containing only carbon and hydrogen.
Its carbon skeleton may contain chains, rings, or multiple bonds.
For example, methane and ethene are both hydrocarbons despite having different bonding structures.
Hydrogen bond
Chemistry · Definition
Hydrogen bonding is a directional interaction involving hydrogen bonded to a sufficiently electronegative atom and an appropriate electron-rich acceptor.
In the introductory model, N, O, and F commonly provide the relevant partners.
For example, water molecules hydrogen bond when one molecule's O–H group interacts with another molecule's oxygen lone pair.
Hydronium
Chemistry · Definition
Hydronium is the H₃O⁺ ion used in the introductory model of proton behavior in water.
Aqueous H⁺ notation is shorthand for a solvated proton rather than a bare proton moving independently in the liquid.
For example, a Brønsted acid can transfer a proton to water and form hydronium.
ICE table
Chemistry · Definition
An ICE table organizes initial concentrations, stoichiometric changes, and equilibrium concentrations.
Its change entries follow the balanced equation and do not imply equal changes for every species.
For example, producing two units of a product for each reactant consumed requires twice the concentration-change entry for that product.
Ideal gas
Physics 2 · Definition
An ideal gas model treats particles as having negligible volume and no sustained intermolecular forces between brief elastic collisions.
The equilibrium state obeys the ideal gas law using absolute pressure and kelvin temperature.
For example, a dilute gas far from condensation often approximates this model better than a dense gas near a phase change.
Ideal gas law
Chemistry · Definition
The ideal gas law relates pressure, volume, amount, and absolute temperature in an idealized gas model.
It treats particles as having negligible volume and no intermolecular attractions.
For example, at fixed amount and temperature, doubling the container volume halves the ideal gas pressure.
Ideal gas law
Physics 2 · Definition
The ideal gas law relates absolute pressure, occupied volume, amount of gas, and kelvin temperature.
The forms using moles and using particle count express the same state relationship with different constants.
For example, heating a fixed amount of ideal gas in a rigid sealed container raises its pressure.
Implicit differentiation
Calculus AB · Calculus BC · Definition
Implicit differentiation finds a derivative from an equation relating variables without first solving explicitly for one variable.
When y depends on x, differentiating a term containing y requires the chain rule and a factor of dy/dx.
For example, x²+y²=1 gives 2x+2y(dy/dx)=0.
Solving that relation for dy/dx gives −x/y only where y≠0 and the relation supports the local function being considered.
Improper integral
Calculus AB · Calculus BC · Definition
An improper integral has an infinite integration interval or an integrand that becomes unbounded within the interval or at an endpoint.
It is defined using limits of proper integrals rather than substituting infinity as a number.
For example, the integral of 1/x² from 1 to infinity converges to 1.
If an interior singularity splits an interval, the separate improper integrals must both converge.
Impulse
Physics 1 · Physics 2 · Definition
Impulse is the change in linear momentum produced by a force over a time interval.
It equals the signed vector area under a force-versus-time graph and is measured in newton-seconds.
For example, a padded landing spreads the stopping impulse over more time, reducing the average force for the same momentum change.
Incompressible fluid
Physics 1 · Physics 2 · Definition
An incompressible-fluid model treats density as constant despite pressure changes in the modeled situation.
This approximation often works for liquids over modest pressure ranges but is not a universal property of all fluids.
For example, water flowing through a narrowing pipe can often be treated as incompressible.
Indefinite integral
Calculus AB · Calculus BC · Definition
An indefinite integral denotes the family of antiderivatives of its integrand on an interval.
The constant of integration records the possible constant differences among those functions.
For example, the indefinite integral of cos x is sin x+C.
It has no pair of numerical bounds and does not by itself give a single accumulated amount.
Indeterminate form
Calculus AB · Calculus BC · Definition
An indeterminate form is a pattern in a proposed limit that does not by itself determine the limit’s value.
The forms 0/0 and infinity divided by infinity can lead to several different outcomes.
For example, x/x approaches 1 as x approaches 0, while x²/x approaches 0 despite the same 0/0 substitution pattern.
The form signals that a justified further analysis is needed.
Index of a sequence or sum
Calculus BC · Definition
An index identifies the position of a term in a sequence or finite or infinite sum.
The starting index determines which term is included first.
For example, a sum indexed by n=0 starts with the term obtained by substituting 0 into its term formula.
Reindexing is valid only when the formula and bounds change together so that the same terms are retained.
Induced dipole
Chemistry · Definition
An induced dipole is a separation of charge produced when an external electric field distorts a particle's electron distribution.
Induction can be caused by an ion, a permanent dipole, or a fluctuating dipole, so the term is broader than London dispersion.
For example, a polar molecule can induce a dipole in a neighboring nonpolar molecule.
Inelastic collision
Physics 1 · Physics 2 · Definition
An inelastic collision does not conserve the colliding system’s total kinetic energy.
Momentum can still be conserved when the net external impulse is negligible, and total energy remains conserved when all stores are included.
For example, carts that latch together undergo a completely inelastic collision and convert some initial kinetic energy into other forms.
Inertia
Physics 1 · Physics 2 · Definition
Inertia is an object’s resistance to a change in its velocity.
Mass measures translational inertia, whereas rotational inertia also depends on how mass is distributed relative to an axis.
For example, a loaded cart accelerates less than an empty cart under the same net force.
Inertial frame
Physics 1 · Physics 2 · Definition
An inertial frame is a reference frame in which an object with no net external force moves at constant velocity.
Frames moving at constant velocity relative to an inertial frame are also inertial in the Newtonian model.
For example, a steadily moving train is approximately inertial, whereas a train accelerating out of a station is not.
Inertial mass
Physics 1 · Physics 2 · Definition
Inertial mass measures how strongly an object resists acceleration under a given net force.
It is the mass in Newton’s second law and is measured in kilograms.
For example, comparing accelerations under the same calibrated force lets an experiment compare two objects’ inertial masses.
Inference for a regression slope
Statistics · Definition
Inference for a regression slope assesses a population linear relationship using the slope fitted to sample data.
It relies on an appropriate design and assumptions about linearity, independent observations, and the response variation around the line.
For example, a t interval can estimate how expected plant height changes per week within the observed age range.
Infinite series
Calculus BC · Definition
An infinite series is the sum of the terms of an infinite sequence, defined through its partial sums.
The series converges when those partial sums approach a finite value.
For example, 1/2+1/4+1/8+⋯ converges to 1.
Terms approaching 0 are necessary for convergence, but that condition alone is insufficient.
Inflection point
Calculus AB · Calculus BC · Definition
An inflection point is a point on a continuous graph where its concavity changes.
A zero or undefined second derivative may identify a candidate but does not establish that change.
For example, x³ changes concavity at 0, while x⁴ has second derivative zero there without changing concavity.
Check the behavior on both sides of the candidate.
Influential observation
Statistics · Definition
An influential observation substantially changes a fitted model when it is removed.
An unusual point is not necessarily influential, and influence depends on its position and the other data.
For example, a point far along the explanatory axis may strongly change the fitted slope.
Initial condition
Calculus AB · Calculus BC · Definition
An initial condition specifies a solution’s value at a particular input.
It can determine the constant in a family of solutions to a differential equation.
For example, y′=2x with y(0)=3 selects y=x²+3.
The solution must satisfy both the equation and the stated condition on a suitable interval.
Instantaneous rate of change
Calculus AB · Calculus BC · Definition
An instantaneous rate of change is the limiting average rate around one input when that limit exists.
For a differentiable function, it is its derivative at that input.
For example, position s(t)=t² meters has instantaneous rate 4 meters per second at t=2 seconds.
The rate’s units are output units per input unit.
Instantaneous velocity
Physics 1 · Physics 2 · Definition
Instantaneous velocity is the rate of change of position at a particular moment.
On a position-versus-time graph, its component is the slope of the tangent at that time.
For example, the velocity just before a ball lands differs from the average velocity of its entire flight.
Integral
Calculus AB · Calculus BC · Definition
Integration can mean accumulating a function over an interval or finding an antiderivative.
A definite integral gives a number, while an indefinite integral represents a family of antiderivatives.
For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4.
The Fundamental Theorem of Calculus connects these two operations under its hypotheses.
Integral test
Calculus BC · Definition
The integral test compares a positive-term series with an improper integral of a matching function.
The function must be continuous, positive, and decreasing for all sufficiently large inputs, with f(n) equal to the series terms.
For example, applying it to 1/n² uses the convergent integral of 1/x² over [1,∞).
The test determines convergence together, but the integral’s value is generally not the series sum.
Integrand
Calculus AB · Calculus BC · Definition
The integrand is the function or expression being integrated.
The differential identifies the variable of integration.
For example, in the integral of x² with respect to x, the integrand is x².
Bounds belong to the integral’s interval rather than to the integrand itself.
Integration by parts
Calculus BC · Definition
Integration by parts rewrites an integral using the derivative product rule in reverse.
It takes the form ∫u dv=uv−∫v du with suitable choices of u and dv.
For example, choosing u=x and dv=eˣ dx turns ∫x eˣ dx into x eˣ−∫eˣ dx.
The choices should simplify the remaining integral, and definite bounds must be carried through consistently.
Intermediate Value Theorem
Calculus AB · Calculus BC · Definition
The Intermediate Value Theorem states that a function continuous on a closed interval takes every value between its endpoint values.
It establishes the existence of an input rather than a formula for that input or a guarantee of uniqueness.
For example, a continuous function with f(0)=−1 and f(2)=3 must equal 0 somewhere between 0 and 2.
The continuity hypothesis must be checked before using the theorem.
Intermolecular force
Chemistry · Definition
Intermolecular forces are interactions between molecules.
They affect properties such as boiling point and solubility and are distinct from the bonds within each molecule.
For example, vaporizing water separates molecules while largely preserving their O–H bonds.
Internal energy
Physics 2 · Definition
Internal energy is energy associated with the microscopic motion and configuration of a system’s constituents.
It excludes the kinetic energy of the whole system’s bulk motion and depends on the thermodynamic state.
For example, a stationary solid can gain internal energy as its atomic vibrations become more energetic.
Internal force
Physics 1 · Physics 2 · Definition
An internal force is an interaction between parts of a chosen system.
Newton’s third-law partners cancel in the total force sum for the whole system, although individual parts can accelerate.
For example, two skaters pushing each other exert internal forces when the two skaters are analyzed together.
Internal resistance
Physics 2 · Definition
Internal resistance models energy dissipation inside a real electrical source.
When the source delivers current, its terminal voltage can be smaller than its emf because of this internal voltage drop.
For example, connecting a heavy load to a battery can lower the voltage measured at its terminals.
Interquartile range
Statistics · Definition
The interquartile range is the third quartile minus the first quartile.
It measures the spread of the middle half of the ordered data.
For example, quartiles of 10 and 18 give an IQR of 8.
Interval
Calculus AB · Calculus BC · Definition
An interval is a set of real numbers containing every number between any two of its members.
Parentheses exclude an endpoint, while square brackets include it.
For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1.
An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
Interval of convergence
Calculus BC · Definition
The interval of convergence is the full set of real inputs where a power series converges.
It consists of the interior determined by its radius plus any convergent endpoints.
For example, the series x−x²/2+x³/3−⋯ has interval (−1,1].
The radius alone cannot determine whether either endpoint is included.
Inverse function
Calculus AB · Calculus BC · Definition
An inverse function reverses another function’s input-output assignment.
A function needs to be one-to-one on its chosen domain for an inverse function to exist.
For example, x³ has inverse the cube-root function, while x² needs a domain restriction such as x≥0.
The notation f⁻¹ denotes an inverse function rather than the reciprocal 1/f.
Inverse trigonometric function
Calculus AB · Calculus BC · Definition
An inverse trigonometric function returns a principal angle from a trigonometric output.
The original trigonometric function is restricted to a one-to-one interval to define that inverse.
For example, arcsin(1/2)=π/6 under the usual range [−π/2,π/2].
The notation sin⁻¹x means arcsin x, whereas the reciprocal of sine is cosecant.
Ion
Chemistry · Definition
An ion is an atom or group of atoms with a net electric charge.
Ordinary ion formation changes electron count without changing the element's proton count.
For example, a sodium atom losing one electron becomes a sodium ion with charge +1.
Ion
Physics 2 · Definition
An ion is an atom or group of atoms with a nonzero net electric charge.
For an atomic ion, the electron count differs from the proton count while the element remains fixed by its protons.
For example, removing an electron from a neutral atom makes a positive ion.
Ion–dipole force
Chemistry · Definition
An ion–dipole force is an interaction between an ion and a polar molecule.
The molecule orients its oppositely charged region toward the ion.
For example, water's oxygen end points toward a dissolved sodium cation.
Ionic bonding
Chemistry · Definition
Ionic bonding is electrostatic attraction among oppositely charged ions.
An ionic solid normally forms an extended lattice rather than separate ion-pair molecules.
For example, sodium chloride consists of a repeating array of sodium and chloride ions.
Ionic strength
Chemistry · Definition
Ionic strength summarizes dissolved-ion concentrations with weighting by the square of each ion's charge number.
It helps describe deviations between ionic activities and concentrations.
For example, at equal concentration a divalent ion contributes more to ionic strength than a monovalent ion.
Ionization energy
Chemistry · Definition
Ionization energy is the energy required to remove an electron from a gaseous atom or ion.
Removing an electron requires energy, and successive removals can reveal a transition from valence to core electrons.
For example, sodium's second ionization requires much more energy than its first because it removes a core electron.
Isobaric process
Physics 2 · Definition
An isobaric process occurs at constant pressure.
The volume and temperature can change, and a moving boundary can allow pressure-volume work.
For example, heating a gas beneath a freely moving loaded piston can increase its volume while its pressure stays approximately constant.
Isolated system
Physics 1 · Physics 2 · Definition
An isolated system exchanges neither energy nor matter with its surroundings.
In a momentum problem, conservation of total momentum specifically requires zero net external impulse over the interval analyzed.
For example, two colliding carts can be treated as momentum-isolated horizontally when the external horizontal impulse is negligible.
Isothermal process
Physics 2 · Definition
An isothermal process occurs at constant temperature.
For a fixed amount of ideal gas, internal energy remains constant even though heat and work can be transferred.
For example, a gas expanding slowly while exchanging energy with a temperature reservoir can remain isothermal.
Isotope
Chemistry · Definition
Isotopes are atoms of the same element with different numbers of neutrons.
They have the same proton number but different masses.
For example, carbon-12 and carbon-13 each have six protons but have six and seven neutrons, respectively.
Isotope
Physics 2 · Definition
Isotopes are atoms of the same element with different numbers of neutrons.
They share an atomic number but have different mass numbers and can differ in nuclear stability.
For example, carbon-12 and carbon-14 are isotopes because both have six protons but different neutron counts.
Joint distribution
Statistics · Definition
A joint distribution describes the combinations of values of two variables and how often they occur.
For categorical data it can be represented by cell counts or proportions in a two-way table.
For example, the distribution of students across grade-level and transportation-mode combinations is a joint distribution.
Joint frequency
Statistics · Definition
A joint frequency is the count in one cell of a two-way table.
It counts individuals who meet both the row category and the column category.
For example, a cell might count students who are in grade 11 and walk to school.
Joint relative frequency
Statistics · Definition
A joint relative frequency divides one cell’s count by the overall table total.
It describes the proportion of all individuals who have both listed characteristics.
For example, 6 grade-11 walkers among 30 students give a joint relative frequency of 20%.
Jump discontinuity
Calculus AB · Calculus BC · Definition
A jump discontinuity occurs when finite left-hand and right-hand limits exist but differ.
The graph approaches different heights from the two sides.
For example, a function equal to 0 for x<1 and 2 for x≥1 has a jump at 1.
Changing its value at 1 does not make the two side limits equal.
Kelvin temperature scale
Physics 2 · Definition
The kelvin scale measures thermodynamic temperature from absolute zero.
A temperature interval of one kelvin has the same size as one degree Celsius, but their zero points differ.
For example, temperature ratios in the ideal gas law must use kelvin values rather than Celsius values.
Ketone
Chemistry · Definition
A ketone contains a carbonyl carbon bonded to two carbon-containing groups.
The carbonyl contributes polarity and can accept hydrogen bonds in the introductory model.
For example, acetone is a ketone with a methyl group on each side of its carbonyl carbon.
Kinematics
Physics 1 · Physics 2 · Definition
Kinematics describes motion using position, velocity, acceleration, and time without first analyzing its forces.
The familiar constant-acceleration equations apply only over intervals in which that acceleration is constant.
For example, a ball falling near Earth with negligible air resistance can be modeled with constant vertical acceleration.
Kinetic energy
Physics 1 · Physics 2 · Definition
Kinetic energy is energy associated with motion relative to a chosen reference frame.
For a nonrelativistic translating object it is one half of its mass times speed squared.
For example, doubling a cart’s speed quadruples its translational kinetic energy when its mass stays fixed.
Kinetic friction
Physics 1 · Physics 2 · Definition
Kinetic friction acts between surfaces that slide relative to one another.
It opposes their relative slipping and is often modeled with magnitude equal to the kinetic-friction coefficient times the normal force.
For example, a rapidly moving belt can exert forward kinetic friction on a slower box sliding on it.
Kinetic molecular theory
Chemistry · Definition
Kinetic molecular theory connects gas pressure and temperature to particle motion and collisions.
In its ideal form, particles move randomly, occupy negligible volume, and undergo elastic collisions without intermolecular attractions.
For example, hotter ideal gas particles have a greater average translational kinetic energy.
Kirchhoff’s junction rule
Physics 2 · Definition
Kirchhoff’s junction rule says total current entering a node equals total current leaving when charge does not accumulate there.
It follows from conservation of charge and applies regardless of how the branches are drawn.
For example, current entering a branching connection divides among the outgoing paths without charge being consumed at the node.
Kirchhoff’s loop rule
Physics 2 · Definition
Kirchhoff’s loop rule sets the signed sum of potential changes around a closed circuit loop to zero in the circuit model used here.
It expresses energy accounting per unit charge and requires consistent signs for sources and components.
For example, a battery’s potential rise balances the total resistor drops around a simple steady loop.
L’Hospital’s rule
Calculus AB · Calculus BC · Definition
L’Hospital’s rule can evaluate certain quotient limits with the indeterminate forms 0/0 or infinity divided by infinity.
Its hypotheses require appropriate differentiability near the target, a nonzero denominator derivative there, and a suitable limit of the derivative quotient.
For example, lim(x→0) sin x/x equals lim(x→0) cos x/1=1.
Differentiating numerator and denominator separately here is a limit theorem, not the quotient rule for derivatives.
Lagrange error bound
Calculus BC · Definition
The Lagrange error bound controls the difference between a sufficiently differentiable function and its degree-n Taylor polynomial.
If the magnitude of the next derivative is at most M between the center c and the input x, the error is at most M|x−c|ⁿ⁺¹/(n+1)!.
For example, approximating eˣ near 0 can use a bound on eᵗ over the interval between 0 and x.
The derivative bound must hold across that whole interval, not merely at the center.
Latent heat
Physics 2 · Definition
Specific latent heat is the energy transferred per unit mass during a specified phase change at the stated conditions.
During an ideal equilibrium phase change at constant pressure, energy can change microscopic arrangement while temperature stays constant.
For example, melting ice requires energy even while an ice–water mixture remains at its melting temperature.
Law of large numbers
Statistics · Definition
The law of large numbers describes how an average or relative frequency approaches its long-run target as repetitions increase under appropriate assumptions.
It does not require short sequences to look balanced or make earlier outcomes force later outcomes.
For example, the proportion of red cards drawn with replacement tends toward one half over many independent draws.
Le Châtelier’s principle
Chemistry · Definition
Le Châtelier’s principle predicts the qualitative response when a system initially at equilibrium is disturbed.
A justified explanation distinguishes the immediate disturbance from the subsequent reaction toward a new equilibrium.
For example, adding a reactant can favor its net consumption, while temperature changes can also alter the equilibrium constant.
Least-squares regression
Statistics · Definition
Least-squares regression fits a line predicting a response from an explanatory variable.
The fitted line minimizes the sum of squared vertical residuals between observed and predicted responses.
For example, a fitted line can predict a plant’s height from its age, within the range of observed ages.
Left-skewed distribution
Statistics · Definition
A left-skewed distribution has a longer tail toward smaller values.
Most observations may lie toward the higher end while a few small values extend the left tail.
For example, scores on an easy test can be left-skewed when most scores are high.
Lenz’s law
Physics 2 · Definition
Lenz’s law gives an induced current direction whose magnetic effect opposes the change in flux that produced it.
It opposes the change rather than necessarily opposing the existing field itself.
For example, when an outward magnetic flux decreases, the induced current can create an outward field that resists that decrease.
Lever arm
Physics 1 · Physics 2 · Definition
A lever arm is the perpendicular distance from a chosen rotation axis to a force’s line of action.
It determines the torque magnitude for a given force and need not equal the distance to the point of application.
For example, a force aimed directly toward a hinge has zero lever arm and produces no torque about that hinge.
Lewis acid and base
Chemistry · Definition
A Lewis acid accepts an electron pair, and a Lewis base donates an electron pair to form a bonding interaction.
This definition includes some acid–base interactions without proton transfer.
For example, a nitrogen lone pair can be donated when ammonia binds to an electron-pair acceptor.
Lewis structure
Chemistry · Definition
A Lewis structure represents valence electrons using bonds and lone pairs.
Its construction requires an electron count, an atom arrangement, and a final check against the total available electrons and charge.
For example, water's Lewis structure has two O–H bonds and two lone pairs on oxygen.
Ligand
Chemistry · Definition
A ligand is an ion or molecule that binds to a central atom or ion in a coordination complex.
In the common introductory model, it donates an electron pair to the central metal.
For example, water molecules can act as ligands around a metal ion.
Limit
Calculus AB · Calculus BC · Definition
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound.
For an input approaching a finite target, its value at that exact input does not determine the limit.
For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1.
A finite two-sided limit requires the left and right approaches to agree.
Limit laws
Calculus AB · Calculus BC · Definition
Limit laws allow limits of sums, products, and suitable quotients to be assembled from the limits of their parts.
The component limits must exist as required, and a quotient law needs a nonzero denominator limit.
For example, if f approaches 2 and g approaches 3, then f+g approaches 5 and fg approaches 6.
Direct substitution into a quotient with denominator limit 0 is not justified by the quotient limit law.
Limiting reactant
Chemistry · Definition
The limiting reactant is consumed first under the specified complete-reaction model and limits the product amount.
Identifying it requires comparing available amounts with the balanced equation's ratios.
For example, abundant hydrogen cannot produce more water after the available oxygen has been consumed.
Linear differential equation
Calculus BC · Definition
A differential equation is linear when the unknown function and its derivatives appear only to the first power and are not multiplied together.
Its coefficients may depend on the independent variable.
For example, y′+xy=x² is linear, while y′+y²=0 is nonlinear.
The sum of solutions is guaranteed to remain a solution for a homogeneous linear equation, but generally not for a nonhomogeneous one.
Linear momentum
Physics 1 · Physics 2 · Definition
Linear momentum is a vector describing an object’s motion and equals mass times velocity in the nonrelativistic model.
The vector sum of momenta is conserved when the net external impulse on the system is zero.
For example, two identical carts moving equally fast in opposite directions have zero total linear momentum but nonzero total kinetic energy.
Linear transformation of a random variable
Statistics · Definition
A linear transformation multiplies a random variable by a constant and adds another constant.
Adding a constant shifts its mean, while multiplying by a constant scales its standard deviation by the constant’s absolute value.
For example, converting a measured length from meters to centimeters multiplies both its mean and standard deviation by 100.
Linearization
Calculus AB · Calculus BC · Definition
A linearization replaces a differentiable function near an input with its tangent line.
At x=a, the approximation is L(x)=f(a)+f′(a)(x−a).
For example, √x near 4 is approximated by 2+(x−4)/4.
The approximation is local, and its error can grow as the input moves away from a.
Local extremum
Calculus AB · Calculus BC · Definition
A local extremum is a value at least as large as, or at least as small as, all values at nearby allowed inputs.
A local maximum compares nearby heights above, while a local minimum compares them below.
For example, x² has a local minimum at 0.
A local extremum need not be the greatest or least value over the entire domain.
Logarithm
Calculus AB · Calculus BC · Definition
A logarithm gives the exponent required to obtain a number from a specified base.
For real logarithms, the argument is positive and the base is positive and different from 1.
For example, log₂8=3 because 2³=8, while ln uses the base e.
Logarithm rules convert products into sums and powers into constant multiples, subject to the domain restrictions.
Logarithmic transformation
Statistics · Definition
A logarithmic transformation replaces positive values by their logarithms.
It can turn an exponential or power relationship into a linear relationship when the appropriate variables are transformed.
For example, plotting the logarithm of bacterial count against time can reveal exponential growth as a line.
Logistic model
Calculus BC · Definition
A logistic model limits positive growth as a quantity approaches a carrying capacity.
A standard form is y′=ky(1−y/K), with k>0 and capacity K>0.
For example, with K=1000, a population rising from 100 to 900 has its relative growth rate y′/y decrease from 0.9k to 0.1k.
The equilibrium solutions y=0 and y=K must be considered separately when solving by separation.
London dispersion forces
Chemistry · Definition
London dispersion forces arise from fluctuating electron distributions and the dipoles they induce in neighboring particles.
They occur between all atoms and molecules, including nonpolar ones.
For example, dispersion attracts neighboring argon atoms even though they have no permanent molecular dipoles.
Lone pair
Chemistry · Definition
A lone pair is a pair of valence electrons assigned to one atom rather than to a bond in a Lewis model.
Lone pairs influence geometry and can participate in interactions or new bonds.
For example, oxygen's two lone pairs help explain water's bent molecular shape.
Longitudinal wave
Physics 2 · Definition
In a longitudinal wave, the disturbance is parallel to the direction of propagation.
Compression and rarefaction can travel while particles oscillate locally along that direction.
For example, sound in air is a longitudinal pressure disturbance rather than a stream of air traveling all the way from source to listener.
Lurking variable
Statistics · Definition
A lurking variable is an unmeasured or unaccounted-for variable that helps explain an observed relationship.
It may create confounding when it is related to both variables being studied.
For example, prior preparation might influence both time spent studying and test score.
Maclaurin series
Calculus BC · Definition
A Maclaurin series is a Taylor series centered at 0.
Its coefficients are determined by the function’s successive derivatives evaluated at 0.
For example, sin x has Maclaurin series x−x³/3!+x⁵/5!−⋯.
As with any Taylor series, equality to the function requires the appropriate convergence and remainder justification.
Magnetic dipole
Physics 2 · Definition
A magnetic dipole has a paired north–south magnetic character, modeled by a small magnet or a current loop.
Its field lines form closed loops, and an external field can exert a turning torque on it.
For example, cutting a bar magnet creates smaller dipoles rather than isolated north and south poles.
Magnetic domain
Physics 2 · Definition
A magnetic domain is a region of a ferromagnetic material in which many microscopic magnetic dipoles are aligned.
Different domains can have different orientations, so their contributions may largely cancel before magnetization.
For example, an applied magnetic field can favor domains aligned with it and leave a piece of iron magnetized.
Magnetic field
Physics 2 · Definition
A magnetic field assigns a vector direction and strength to each location and predicts interactions with moving charges, currents, and magnetic materials.
Its strength is measured in teslas, and the force on a moving charge need not point along the field.
For example, a compass aligns with the local field while a moving charge can be deflected perpendicular to it.
Magnetic flux
Physics 2 · Definition
Magnetic flux measures the magnetic field’s perpendicular contribution through a chosen surface.
For a uniform field and flat surface it depends on field strength, area, and the angle to the area normal, and it is measured in webers.
For example, rotating a loop from face-on to edge-on to a uniform field reduces its flux magnitude toward zero.
Magnetic permeability
Physics 2 · Definition
Magnetic permeability relates a material’s magnetic response to an applied field in a specified model.
The response can depend on field strength, direction, and temperature, so one constant need not describe every condition.
For example, a ferromagnetic core approaching saturation does not keep increasing its magnetization in direct proportion to the applied field.
Magnitude
Physics 1 · Physics 2 · Definition
Magnitude is the nonnegative size of a vector quantity.
It does not specify direction, and it differs from a signed component.
For example, a westward velocity can have a negative x component while its speed, the velocity magnitude, stays positive.
Margin of error
Statistics · Definition
A margin of error is the amount added to and subtracted from a point estimate in a symmetric confidence interval.
In the procedures here it combines a critical value with an estimated standard error and addresses modeled sampling variability.
For example, an estimate of 40% with a margin of 3 percentage points gives an interval from 37% to 43%.
Marginal distribution
Statistics · Definition
A marginal distribution describes one variable on its own after the other variable is summed or otherwise accounted for.
In a two-way table, it uses row or column totals, expressed as counts or proportions.
For example, adding all grade-level counts for each transportation mode gives the marginal distribution of transportation mode.
Marginal frequency
Statistics · Definition
A marginal frequency is a row or column total in a two-way table.
It counts one category without separating individuals by the other variable.
For example, the total number of walkers includes walkers from every grade level.
Marginal relative frequency
Statistics · Definition
A marginal relative frequency divides a row or column total by the overall table total.
The collection of these proportions describes one categorical variable on its own.
For example, 12 walkers among 30 students give a marginal relative frequency of 40%.
Mass defect
Physics 2 · Definition
Mass defect is the difference between the summed rest masses of separated constituents and the rest mass of their bound system.
For a nucleus, the constituents include all its free protons and neutrons.
For example, a positive nuclear mass defect corresponds to energy released when those nucleons assemble into the bound nucleus.
Mass density
Physics 1 · Physics 2 · Definition
Mass density is mass per unit volume of a material or sample.
It is commonly measured in kilograms per cubic meter and can vary with temperature or pressure.
For example, equal-volume samples of water and oil generally have different masses because their densities differ.
Mass fraction
Chemistry · Definition
Mass fraction is a component's mass divided by the total mass of the sample.
Multiplying that fraction by 100 gives its mass percentage.
For example, 5 grams of a solute in 100 grams of solution corresponds to a mass fraction of 0.05 and a mass percentage of 5 percent.
Mass number
Physics 2 · Definition
Mass number is the total number of protons and neutrons in a nucleus.
It is a count rather than the atom’s measured mass in kilograms or atomic mass units.
For example, subtracting atomic number from mass number gives the neutron count of an isotope.
Mass spectrum
Chemistry · Definition
A mass spectrum displays detected ions according to their mass-to-charge ratio and relative signal.
In this lesson's elemental model, singly charged monatomic ions make peak positions correspond numerically to isotope masses.
For example, two isotope peaks can be combined with their relative abundances to determine a weighted average atomic mass.
Matched pairs design
Statistics · Definition
A matched pairs design compares two treatments within matched units or within the same unit.
Assignment within pairs or treatment order is randomized when the design permits, and analysis uses the pairwise differences.
For example, two shoe designs can be compared on the same person with their left-right assignment randomized.
Maxwell–Boltzmann distribution
Physics 2 · Definition
A Maxwell–Boltzmann speed distribution describes the range of particle speeds in a classical gas at thermal equilibrium.
For the same species, a higher temperature shifts the distribution toward larger speeds and broadens it while preserving the normalized total population.
For example, the tallest point identifies the most likely speed rather than the speed of every particle or the rms speed.
Mean
Statistics · Definition
The mean is a measure of center often called an average.
For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists.
Means can be pulled toward extreme values and are not generally resistant.
For example, the arithmetic mean of 2, 4, and 9 is 5.
Mean Value Theorem
Calculus AB · Calculus BC · Definition
The Mean Value Theorem connects an average rate on a closed interval to an instantaneous rate at an interior point.
It requires continuity on the closed interval and differentiability on its interior.
For example, f(x)=x² on [1,3] has average slope 4 and derivative 4 at x=2.
The theorem guarantees at least one such interior point but need not identify a unique one.
Mechanical energy
Physics 1 · Physics 2 · Definition
Mechanical energy is the sum of the kinetic and potential energies included in a model.
It stays constant when transfers to or from other energy stores and across the chosen boundary are absent.
For example, a pendulum’s mechanical energy can remain approximately constant while energy shifts between motion and gravitational configuration.
Mechanical equilibrium
Physics 1 · Physics 2 · Definition
Mechanical equilibrium means there is no translational or angular acceleration for the system being considered.
For a rigid body, both the net external force and the net external torque must vanish.
For example, a stationary ladder supported against a wall requires both force balance and torque balance.
Mechanical work
Physics 1 · Physics 2 · Definition
Mechanical work is energy transferred by a force through displacement of its point of application.
For a constant force on a particle, it depends on the force component along the particle’s displacement, and it can be positive, negative, or zero.
For example, a magnetic force perpendicular to a particle’s displacement does zero work on that particle.
Median
Statistics · Definition
The median is the middle of an ordered numerical data set.
For an even number of observations, it is the average of the two middle values under the usual convention.
For example, the median of 2, 4, and 20 is 4.
Metallic bonding
Chemistry · Definition
Metallic bonding involves metal centers held together by electrons delocalized through the solid.
The mobile electronic charge helps explain electrical conductivity and the ability of many metals to deform.
For example, copper remains bonded as its atoms shift when a wire is bent.
Midpoint rule
Calculus AB · Calculus BC · Definition
The midpoint rule approximates a definite integral by sampling at the center of each subinterval.
Each sampled height is multiplied by its own subinterval width.
For example, one midpoint rectangle for x² on [0,2] has height 1 and width 2, giving 2.
For a concave-up function the midpoint sum underestimates the integral, while for a concave-down function it overestimates it.
Mixture
Chemistry · Definition
A mixture contains two or more substances whose proportions can vary.
Its components retain their chemical identities until a chemical reaction changes them.
For example, water and ethanol can form mixtures with many different compositions.
Mobile phase
Chemistry · Definition
The mobile phase is the fluid that carries a sample through a chromatographic system.
A component's movement depends on its interactions with this phase and the stationary phase.
For example, a solvent can carry dissolved dyes along a chromatographic strip.
Mode
Statistics · Definition
A mode is a value occurring with the greatest frequency in a data set.
For a density or grouped graph, a mode refers to a location or interval at a peak.
For example, the mode of 2, 2, 3, and 5 is 2.
Molality
Chemistry · Definition
Molality is the amount of solute in moles divided by the solvent mass in kilograms.
It uses solvent mass rather than solution volume and differs from molarity.
For example, one mole of solute dissolved in two kilograms of water has molality 0.50 mole per kilogram.
Molar absorptivity
Chemistry · Definition
Molar absorptivity describes a species' absorption strength at a specified wavelength in the Beer–Lambert model.
It depends on the absorbing species and measurement conditions, while concentration and path length are separate factors.
For example, selecting a wavelength near an absorption maximum can give a larger absorbance for the same concentration and path length.
Molar mass
Chemistry · Definition
Molar mass is the mass of one mole of a specified substance.
It is commonly expressed in grams per mole and connects a sample's mass to its amount of substance.
For example, water has a molar mass of approximately 18.02 grams per mole.
Molarity
Chemistry · Definition
Molarity is the amount of a solute in moles divided by the final solution volume in liters.
It describes concentration rather than the total amount of solute.
For example, 0.10 mole dissolved to make 0.50 liter of solution gives a molarity of 0.20 mole per liter.
Mole
Chemistry · Definition
A mole is an amount of substance containing exactly 6.02214076 × 10²³ specified entities.
The entities can be atoms, molecules, ions, or formula units, so the counted entity must be named.
For example, one mole of oxygen molecules contains two moles of oxygen atoms.
Mole
Physics 2 · Definition
A mole is an amount of substance containing Avogadro’s number of specified entities.
Those entities must be named, such as atoms, molecules, or ions, and a mole is not itself a mass unit.
For example, one mole of helium atoms and one mole of oxygen molecules contain the same number of specified entities but have different masses.
Mole fraction
Chemistry · Definition
Mole fraction is a component's amount in moles divided by the mixture's total amount in moles.
The fractions of all components sum to one.
For example, one mole of one gas mixed with three moles of another gives the first gas a mole fraction of 0.25.
Molecular formula
Chemistry · Definition
A molecular formula gives the actual number of each kind of atom in a molecule.
It can be a whole-number multiple of the empirical formula.
For example, glucose has molecular formula C₆H₁₂O₆ and empirical formula CH₂O.
Molecular orbital
Chemistry · Definition
A molecular orbital is an electronic state described over a molecule rather than assigned exclusively to a single atom.
Molecular-orbital models can describe bonding, antibonding, and nonbonding electron arrangements.
For example, occupying a bonding orbital between two atoms can contribute to holding those atoms together.
Molecular polarity
Chemistry · Definition
Molecular polarity concerns a molecule's overall separation of charge.
Both bond polarity and molecular geometry determine whether individual bond dipoles cancel.
For example, carbon dioxide has polar bonds but is nonpolar overall because its linear bond dipoles cancel.
Multimodal distribution
Statistics · Definition
A multimodal distribution has several main peaks.
Apparent peaks can depend on how a histogram groups observations into intervals.
For example, data collected from several distinct age groups may have several peaks.
Mutually exclusive events
Statistics · Definition
Mutually exclusive events cannot occur together in the same trial.
Their intersection is empty, so the probability that either occurs is the sum of their probabilities.
For example, rolling exactly 1 and rolling exactly 4 are mutually exclusive outcomes of one die roll.
Negative association
Statistics · Definition
Two variables have a negative association when larger values of one tend to accompany smaller values of the other.
The tendency can be linear or curved and does not by itself show cause and effect.
For example, higher fuel efficiency may accompany lower fuel use per mile.
Nernst equation
Chemistry · Definition
The Nernst equation relates a cell's potential to temperature and the reaction quotient for its written reaction.
It distinguishes the current composition-dependent potential from the standard reference potential.
For example, approaching chemical equilibrium brings the ideal cell's actual potential toward zero.
Net change
Calculus AB · Calculus BC · Definition
Net change is the final amount minus the initial amount.
Integrating a rate gives this signed change when the model and integral are valid.
For example, moving 5 meters right and then 2 meters left gives net displacement 3 meters.
Opposite contributions cancel, so net change can differ from the total amount of movement or activity.
Net force
Physics 1 · Physics 2 · Definition
Net force is the vector sum of all forces acting on the object or system under consideration.
For constant mass in an inertial frame, it equals mass times acceleration.
For example, equal upward and downward forces give zero net force even though neither force is zero.
Net ionic equation
Chemistry · Definition
A net ionic equation shows the species that undergo the chemical change.
It removes spectator ions while preserving atom and charge balance.
For example, precipitation of silver chloride can be written using aqueous silver and chloride ions forming solid silver chloride.
Neutrino
Physics 2 · Definition
Neutrinos are electrically neutral elementary particles with very small masses and weak interactions with ordinary matter.
Neutrinos and antineutrinos carry energy, momentum, and lepton number in beta-decay accounting.
For example, an electron antineutrino accompanies the electron emitted in beta-minus decay.
Newton’s method
Calculus AB · Definition
Newton’s method estimates a root by repeatedly replacing the current input with the horizontal intercept of its tangent line.
The update is xₙ₊₁=xₙ−f(xₙ)/f′(xₙ) where the denominator is nonzero.
For example, starting at 2 for f(x)=x²−2 produces the next estimate 1.5.
A poor starting value or problematic derivative can cause failure, so convergence is not guaranteed.
Nitrile
Chemistry · Definition
A nitrile contains a carbon atom triple-bonded to nitrogen.
The nitrile carbon is part of the functional group and affects systematic naming.
For example, acetonitrile contains a methyl group attached to a nitrile carbon.
Nonconservative force
Physics 1 · Physics 2 · Definition
A nonconservative force can do work that depends on the path followed.
Its effects cannot be described by a single configuration-dependent potential energy for that interaction alone.
For example, sliding friction transfers more mechanical energy into thermal energy along a longer sliding path under the same conditions.
Nonresponse
Statistics · Definition
Nonresponse occurs when selected individuals do not supply usable responses.
It can cause bias if nonrespondents differ systematically from respondents on the study’s variables.
For example, selected students with long commutes may be less likely to answer a transportation survey.
Nonsampling error
Statistics · Definition
Nonsampling errors arise from problems other than selecting a finite sample.
They include inaccurate answers, missed individuals, recording mistakes, and faulty measurement.
For example, entering a recorded age incorrectly can affect a census as well as a sample survey.
Normal distribution
Statistics · Definition
A Normal distribution is a continuous, symmetric bell-shaped distribution determined by its mean and standard deviation.
Many symmetric distributions are not Normal, so symmetry by itself is insufficient.
For example, a Normal model centered at 50 with standard deviation 5 places most values relatively close to 50.
Normal distribution in integration
Calculus BC · Definition
A normal distribution is a continuous bell-shaped probability model determined by a mean and a positive standard deviation.
The standard normal model has mean 0 and standard deviation 1.
For example, the probability that a standard normal variable lies between 0 and 1 is the integral of its density over that interval.
The full density has total area 1, and interval areas rather than single-point heights give probabilities.
Normal force
Physics 1 · Physics 2 · Definition
A normal force is a contact force perpendicular to the local surface of contact.
Its magnitude is set by the motion and the other forces, so it need not equal the object’s weight.
For example, a wall’s normal force on a leaning ladder points horizontally rather than upward.
Normal probability plot
Statistics · Definition
A Normal probability plot compares ordered observations with the pattern expected from a Normal distribution.
A roughly straight pattern supports the approximation, while strong curvature or isolated points can reveal departures.
For example, a residual Normal probability plot can help assess a regression model’s Normal-error assumption.
nth-term test for divergence
Calculus BC · Definition
The nth-term test shows that a series diverges if its terms fail to approach 0.
If the term limit is nonzero or does not exist, a finite sum is impossible.
For example, the series with terms n/(n+1) diverges because its terms approach 1.
When the term limit is 0, the test is inconclusive.
Nuclear binding energy
Physics 2 · Definition
Nuclear binding energy is the energy required to separate a nucleus into free protons and neutrons.
It equals the mass defect multiplied by the square of the speed of light, with both proton and neutron masses included.
For example, forming a bound nucleus releases energy because its rest mass is smaller than that of its separated nucleons.
Nuclear fission
Physics 2 · Definition
Nuclear fission splits a nucleus into smaller nuclei and may also emit other particles.
The energy outcome depends on the reaction’s mass balance, so not every arbitrary splitting releases energy.
For example, fission of an appropriate heavy nucleus can release kinetic energy, photons, and neutrons that can initiate further reactions.
Nuclear fusion
Physics 2 · Definition
Nuclear fusion combines smaller nuclei into a larger nucleus and can also emit other particles.
Energy release depends on the reaction’s mass balance, and initially the charged nuclei must come close despite electric repulsion.
For example, suitable light-hydrogen-isotope reactions can release energy when their products are more tightly bound.
Null hypothesis
Statistics · Definition
A null hypothesis specifies the reference claim or model used to calculate a test’s probabilities.
It commonly states no population difference or a particular parameter value.
For example, a null hypothesis can state that the population mean travel time equals 15 minutes.
Observation
Statistics · Definition
An observation is a recorded value or set of values for an individual or occasion.
The individual may be a person, object, or event rather than a human subject.
For example, one plant’s measured height is an observation in a plant-growth study.
Observational study
Statistics · Definition
An observational study measures individuals without assigning the treatments or conditions being compared.
It can reveal associations, but confounding can prevent a causal conclusion.
For example, recording students’ existing sleep habits and scores is observational when researchers do not assign sleep schedules.
Observed count
Statistics · Definition
An observed count is the actual number of sampled individuals in a category or table cell.
It is compared with the expected count to measure disagreement with a null model.
For example, if 13 sampled students chose walking, the observed count for walking is 13.
Octet rule
Chemistry · Definition
The octet rule is a useful model in which many main-group atoms form arrangements with eight valence-shell electrons.
It has important exceptions, including hydrogen's duet and electron-deficient or odd-electron species.
For example, methane's Lewis structure gives carbon an octet through four bonds.
Ohm’s law
Physics 2 · Definition
Ohm’s law describes an element whose current is proportional to its potential difference under the stated conditions.
Such an element has constant resistance over the modeled range, giving a straight-line current–voltage relationship through the origin.
For example, a fixed-temperature ideal resistor follows this model, whereas a warming filament bulb need not.
One-sample t procedure
Statistics · Definition
A one-sample t procedure makes inference about a population mean using the sample mean and sample standard deviation.
It requires an appropriate study design and a sufficiently suitable distribution of the mean, with particular care for small samples, strong skewness, and outliers.
For example, a random sample of travel times can estimate a population mean when the assumptions are defensible.
One-sample z procedure for a proportion
Statistics · Definition
A one-sample z procedure uses a Normal approximation for inference about one population proportion.
A test uses the hypothesized proportion in its null standard error, while an interval usually estimates the standard error from the sample proportion.
For example, a random sample can be used to test a stated population preference proportion after checking the procedure’s conditions.
One-sided limit
Calculus AB · Calculus BC · Definition
A one-sided limit uses inputs approaching a target from just one side.
A left-hand limit uses smaller inputs, and a right-hand limit uses larger inputs.
For example, a function that equals 0 left of 2 and 1 right of 2 has different one-sided limits at 2.
The two-sided limit exists as a finite real number only when both one-sided limits exist and agree.
One-sided test
Statistics · Definition
A one-sided test seeks evidence for a specified direction of departure from the null claim.
Its relevant extreme outcomes lie in one direction of the test statistic’s distribution.
For example, a greater-than alternative asks whether a population mean exceeds a stated value.
One-to-one function
Calculus AB · Calculus BC · Definition
A function is one-to-one when different inputs always have different outputs.
Equivalently, each horizontal line meets its graph at most once.
For example, x³ is one-to-one on the real numbers, whereas x² is not because 2 and −2 have the same output.
This property allows a function’s inverse relation to be a function.
One-way table
Statistics · Definition
A one-way table lists counts or proportions for the categories of one variable.
The categories should cover the intended outcomes without overlap.
For example, a table listing counts for walking, cycling, and riding describes one transportation variable.
Optical magnification
Physics 2 · Definition
Optical magnification is the ratio of image height to object height in the chosen signed convention.
Its magnitude gives the size ratio, while its sign indicates upright or inverted orientation.
For example, an inverted image half the object’s height has negative magnification with magnitude one half.
Optical path length
Physics 2 · Definition
Optical path length accounts for wave phase accumulated along a path through a medium.
For a uniform medium it equals refractive index times geometric path length at the relevant frequency.
For example, equal geometric distances through air and glass generally correspond to different optical path lengths.
Optimization
Calculus AB · Calculus BC · Definition
Optimization finds the greatest or least value of an objective while satisfying the problem’s constraints.
Express the objective in suitable variables, determine the allowed domain, and compare all relevant candidates.
For example, among rectangles with a fixed perimeter, writing area in terms of one side reduces the search to a one-variable problem.
Endpoint and feasibility checks are part of the argument, not optional final details.
Orbital hybridization
Chemistry · Definition
Hybridization is a localized bonding model that combines atomic-orbital descriptions on an atom.
It can help describe bond directions but is not a statement that isolated atomic orbitals physically mix as a timed step.
For example, an sp³ model describes four tetrahedrally directed orbitals around carbon in methane.
Orbital period
Physics 1 · Physics 2 · Definition
An orbital period is the time needed for one complete orbit.
In a two-body model it depends on the attracting mass and the orbit’s size rather than just the orbiting object’s mass.
For example, a larger circular orbit around the same planet has a longer orbital period.
Ordinary differential equation
Calculus BC · Definition
An ordinary differential equation uses derivatives of unknown functions with respect to one independent variable.
Its order is the highest derivative appearing in the equation.
For example, y″+y=0 is an ordinary differential equation of order 2.
Equations involving partial derivatives with respect to several independent variables belong to a different class.
Organic compound
Chemistry · Definition
Organic compounds comprise a broad class of carbon-based compounds, often built around carbon–carbon and carbon–hydrogen structures.
Some carbon-containing substances, such as carbon dioxide and carbonates, are conventionally classified as inorganic.
For example, ethanol is organic while sodium carbonate is usually classified as inorganic.
Oscillation period
Physics 1 · Physics 2 · Definition
An oscillation period is the time required for one complete repeating cycle.
It is the reciprocal of frequency and is measured in seconds.
For example, measuring the time for several pendulum swings and dividing by the number of cycles estimates the period.
Oscillation phase
Physics 1 · Physics 2 · Definition
Phase specifies the position within a repeating oscillation cycle.
A phase difference tells how much one oscillation leads or lags another at a chosen place and time.
For example, two sinusoidal waves half a cycle out of phase can cancel when their amplitudes are equal.
Outlier
Statistics · Definition
An outlier is an observation unusually far from the overall pattern.
The 1.5-IQR rule flags possible outliers below the first quartile minus 1.5 IQR or above the third quartile plus 1.5 IQR.
For example, a very long commute may be flagged for investigation without automatically being a recording error.
Oxidation
Chemistry · Definition
Oxidation is electron loss, represented by an increase in oxidation number.
It occurs together with reduction so that electrons are conserved in the overall reaction.
For example, a zinc atom becoming a zinc ion with charge +2 undergoes oxidation.
Oxidation number
Chemistry · Definition
An oxidation number is a formal electron-accounting value assigned using specified rules.
Changes in oxidation number help identify oxidation and reduction but are not always actual atomic charges.
For example, oxygen usually has oxidation number −2 in water while hydrogen has +1.
Oxidizing agent
Chemistry · Definition
An oxidizing agent causes another species to lose electrons.
The agent accepts those electrons and is itself reduced.
For example, a copper ion acts as an oxidizing agent when it accepts electrons from zinc metal.
p-series
Calculus BC · Definition
A p-series has terms 1/nᵖ for positive integers n.
It converges exactly when p>1.
For example, the p-series with p=2 converges, while p=1 gives the divergent harmonic series.
The boundary p=1 belongs to the divergent cases.
P-value
Statistics · Definition
A p-value is the probability, under the null model and its assumptions, of results at least as extreme as those observed in the direction or directions specified by the test.
It does not give the probability that the null hypothesis is true or measure the practical size of an effect.
For example, a p-value of 0.03 describes a 3% tail probability under the null model, rather than a 3% probability that the null is correct.
Paired t procedure
Statistics · Definition
A paired t procedure analyzes the mean of differences computed within matched pairs.
It uses one-sample t inference on those differences and requires independence across pairs rather than within each pair.
For example, subtracting each student’s before score from the same student’s after score preserves the pairing.
Parallel connection
Physics 2 · Definition
A parallel connection joins elements across the same two nodes.
They therefore have the same potential difference, while their branch currents can differ.
For example, two different resistors connected in parallel across a battery generally carry different currents.
Parallel-axis theorem
Physics 1 · Physics 2 · Definition
The parallel-axis theorem relates rotational inertia about a center-of-mass axis to inertia about a parallel displaced axis.
The displaced-axis value is the center-of-mass value plus total mass times the squared separation of the axes.
For example, a uniform rod has greater rotational inertia about one end than about a parallel axis through its middle.
Paramagnetism
Chemistry · Definition
Paramagnetism is attraction to an applied magnetic field associated with unpaired electrons in the introductory model.
The relevant electron arrangement must be considered rather than inferred from atom count alone.
For example, molecular oxygen has unpaired electrons and is paramagnetic.
Paramagnetism
Physics 2 · Definition
Paramagnetism is a usually weak tendency for a material’s magnetic dipoles to align with an applied field.
The additional alignment generally disappears when the field is removed.
For example, aluminum has a weak positive magnetic response under the conditions described in the course model.
Parametric equations
Calculus BC · Definition
Parametric equations describe coordinates as functions of a shared parameter.
For x=x(t) and y=y(t), the point (x(t),y(t)) traces a curve as t varies.
For example, x=cos t and y=sin t trace the unit circle.
The slope dy/dx equals y′(t)/x′(t) where x′(t)≠0, and the parameter also records orientation and repeated tracing.
Partial charge
Chemistry · Definition
A partial charge describes an uneven distribution of electron density within a bond or molecule.
It is marked by a delta symbol and does not necessarily represent a full ionic charge.
For example, oxygen carries a partial negative charge in water while its hydrogens carry partial positive charges.
Partial derivative
Calculus BC · Definition
A partial derivative measures how a multivariable function changes with one input while the other inputs are held constant.
The chosen variable must be specified.
For example, for f(x,y)=x²y, the partial derivative with respect to x is 2xy.
This is different from differentiating y as an unknown function of x in implicit differentiation.
Partial fractions
Calculus BC · Definition
Partial fraction decomposition writes a rational expression as a sum of simpler fractions.
If the numerator’s degree is at least the denominator’s, polynomial division comes first.
For example, 1/[x(x+1)] equals 1/x−1/(x+1) wherever the original expression is defined.
The factor types and any repeated factors determine the form of the decomposition.
Partial pressure
Chemistry · Definition
A gas's partial pressure is the pressure it would exert alone in the mixture's volume at the same temperature.
In an ideal mixture, the component partial pressures add to the total pressure.
For example, a component with mole fraction 0.20 contributes 0.20 of the total ideal pressure.
Partial sum
Calculus BC · Definition
A partial sum adds the first finitely many terms of a series.
The nth partial sum is Sₙ=a₁+a₂+⋯+aₙ when indexing begins at 1.
For example, the first three terms of 1/2+1/4+1/8+⋯ give partial sum 7/8.
Convergence of a series concerns the limit of these sums, not just the limit of the individual terms.
Particular solution
Calculus AB · Calculus BC · Definition
A particular solution is one specific function satisfying a differential equation.
Initial conditions often select it from a general family.
For example, y=x²+3 is the solution of y′=2x with y(0)=3.
Verify the selected function in both the differential equation and the initial condition.
Passivation
Chemistry · Definition
Passivation occurs when a surface layer reduces further reaction of the underlying material.
It can make an idealized redox prediction differ from the observed reaction rate or cell behavior.
For example, a protective oxide layer can slow a metal's further oxidation.
Pauli exclusion principle
Chemistry · Definition
The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers.
Consequently, an orbital can contain at most two electrons, and those electrons must have opposite spins.
For example, the two electrons in helium's 1s orbital are represented by opposing arrows.
Pendulum
Physics 1 · Physics 2 · Definition
A pendulum is a system that swings about a support under a restoring torque.
A simple pendulum approximates a small mass on a light string and follows simple harmonic motion only for sufficiently small angles.
For example, a short string pendulum completes small swings more quickly than a longer one under the same gravitational field.
Percent ionization
Chemistry · Definition
Percent ionization is the fraction of an initially supplied acid or base that forms the relevant ions, multiplied by 100.
It depends on concentration and conditions as well as the species' equilibrium constant.
For example, a weak acid can have a larger ionized fraction after dilution even though its hydronium concentration decreases.
Percentile
Statistics · Definition
A percentile identifies a location in an ordered distribution by the proportion below or at that location.
Ties and finite samples require a stated convention, so methods can produce slightly different percentile values.
For example, the 80th percentile marks a value with roughly 80% of observations at or below it.
Period of a function
Calculus AB · Calculus BC · Definition
A positive period of a function is an input shift that leaves all its outputs unchanged.
The fundamental period is the smallest such positive shift when one exists.
For example, sin x has fundamental period 2π, while sin(2x) has fundamental period π.
The period describes repetition along the input axis rather than the height of an oscillation.
Permittivity
Physics 2 · Definition
Permittivity describes a medium’s electric response and relates electric fields to charge in the chosen model.
A material’s polarization can make its response differ from that of vacuum.
For example, placing a dielectric between capacitor plates can change the field and capacitance because the material polarizes.
pH
Chemistry · Definition
pH is the negative base-ten logarithm of hydronium activity.
In suitable dilute models, activity is approximated by molar concentration relative to the standard concentration.
For example, lowering hydronium concentration by a factor of ten raises pH by one unit.
Phase change
Chemistry · Definition
A phase change alters a substance's physical state without changing its chemical identity.
It involves a change in particle arrangement and interactions and can transfer energy without raising temperature during a pure-substance transition at fixed pressure.
For example, vaporizing liquid water separates water molecules into a gas.
Phase shift
Calculus AB · Calculus BC · Definition
A phase shift moves a periodic graph horizontally relative to a chosen reference graph.
Writing y=A sin(B(x−h))+D makes the shift h visible when B≠0.
For example, sin(x−π/2) is the sine graph shifted π/2 units to the right.
For a form such as sin(Bx+C), factor B before reading the shift as −C/B.
Photoelectric effect
Physics 2 · Definition
The photoelectric effect is electron emission from a material after it absorbs electromagnetic radiation with sufficient energy.
In the ordinary single-photon model, the photon must meet the material’s work-function threshold before an electron can escape.
For example, making below-threshold light brighter does not cause emission in that model because each photon still has insufficient energy.
Photoelectron spectroscopy
Chemistry · Definition
Photoelectron spectroscopy uses photons to remove electrons and measure their binding energies.
Peak position indicates how tightly electrons are bound, while peak intensity can indicate the relative population of a subshell in the simplified model.
For example, an inner-shell electron normally produces a peak at higher binding energy than a valence electron.
Photon
Chemistry · Definition
A photon is a quantum of electromagnetic radiation.
Its energy increases with frequency and decreases as wavelength increases.
For example, a blue-light photon carries more energy than a red-light photon.
Photon
Physics 2 · Definition
A photon is a quantum of electromagnetic radiation with energy proportional to its frequency.
It has zero rest mass and no electric charge but carries energy and momentum.
For example, increasing light’s frequency increases the energy of each photon, while increasing the number of same-frequency photons increases the total energy.
Physical system
Physics 1 · Physics 2 · Definition
A physical system is the object or collection of objects chosen for an analysis.
Its boundary determines which forces and energy transfers count as internal or external.
For example, friction between a block and a floor is an external interaction for the block alone but internal when both are included.
Pi bond
Chemistry · Definition
A pi bond has bonding electron density on opposite sides of the line joining two nuclei.
In a localized model, it results from sideways overlap of appropriately oriented orbitals.
For example, a carbon–carbon double bond contains one sigma bond and one pi bond.
Pie chart
Statistics · Definition
A pie chart displays category proportions as slices of a circle.
A slice’s angle is proportional to its category’s share of the whole.
For example, a category containing one quarter of the observations occupies one quarter of the circle.
pKa
Chemistry · Definition
pKa is the negative base-ten logarithm of an acid's dissociation constant.
A smaller pKa means a stronger acid under comparable conditions.
For example, when an acid and its conjugate base have equal activities, their buffer's pH equals that acid's pKa.
Placebo
Statistics · Definition
A placebo resembles an active treatment but lacks the active intervention under study.
It helps provide a comparison for effects associated with receiving treatment and expectations.
For example, an inactive pill can be made to look like the experimental pill.
Placebo effect
Statistics · Definition
A placebo effect is a response associated with expectations or the experience of receiving a treatment.
Observed change in a placebo group can also include natural recovery or other influences, so it is not all automatically a placebo effect.
For example, reported discomfort may change after a participant receives an inactive pill.
Planck’s constant
Physics 2 · Definition
Planck’s constant relates the energy of a photon to its frequency.
It also sets the scale connecting particle momentum to de Broglie wavelength.
For example, two photons of different frequencies have energies in the same ratio as their frequencies because the constant is the same for both.
pOH
Chemistry · Definition
pOH is the negative base-ten logarithm of hydroxide activity.
At a specified temperature, pH plus pOH equals the corresponding pKw, which is approximately 14 at 25 °C.
For example, increasing hydroxide activity tenfold lowers pOH by one unit.
Point estimate
Statistics · Definition
A point estimate is the specific numerical value obtained from an estimator for the observed sample.
It gives one estimate without expressing uncertainty by itself.
For example, a sample mean of 14 minutes is a point estimate of the population’s mean travel time.
Point estimator
Statistics · Definition
A point estimator is a rule that uses sample data to produce one estimate of a population parameter.
The rule has a sampling distribution before the sample is observed.
For example, computing the sample mean is a point estimator of the population mean.
Polar coordinates
Calculus BC · Definition
Polar coordinates describe a plane point by a directed radial distance r and angle θ.
The corresponding rectangular coordinates are x=r cos θ and y=r sin θ.
For example, (r,θ)=(2,π/2) describes the point (0,2).
Angles differing by a full turn and suitable negative-radius descriptions can name the same point, so tracing a polar curve needs care.
Polarizability
Chemistry · Definition
Polarizability describes how readily a particle's electron distribution can be distorted.
Larger, more diffuse electron clouds are often more polarizable, strengthening dispersion interactions when other factors are comparable.
For example, iodine is more polarizable than fluorine.
Polymer
Chemistry · Definition
A polymer is a large molecule built from many linked repeating units.
Chain length, branching, and interactions between chains influence its material properties.
For example, polyethylene contains long chains of repeating carbon-based units.
Polynomial
Calculus AB · Calculus BC · Definition
A polynomial in one variable is a finite sum of constant multiples of nonnegative integer powers of that variable.
The degree is the highest exponent with a nonzero coefficient.
For example, 2x³−x+5 is a polynomial of degree 3.
Expressions containing 1/x or √x are not polynomials in x.
Polyprotic acid
Chemistry · Definition
A polyprotic acid can donate more than one proton per molecule in successive steps.
Each step has its own equilibrium constant, so the protons need not dissociate to the same extent.
For example, phosphoric acid can lose three protons through separate equilibria.
Pooled sample proportion
Statistics · Definition
A pooled sample proportion combines successes and sample sizes from two groups.
It estimates a common proportion when a two-proportion null hypothesis states that the population proportions are equal.
For example, 12 successes in 30 trials and 18 in 40 trials give a pooled proportion of 30 divided by 70.
Population
Statistics · Definition
A population is the full set of individuals or measurements a study aims to describe.
It must be defined clearly before deciding whether a sample supports generalization.
For example, all students enrolled in a particular school this year form a population.
Population parameter
Statistics · Definition
A population parameter is a numerical feature of the population or probability model being studied.
It is fixed for that defined population, although its value may be unknown.
For example, the true proportion of all enrolled students who walk to school is a parameter.
Population proportion
Statistics · Definition
A population proportion is the fraction of the entire population with a specified characteristic.
It is a parameter, commonly written p, that a sample proportion can estimate.
For example, the fraction of all enrolled students who walk to school is the population proportion of walkers.
Positive association
Statistics · Definition
Two variables have a positive association when larger values of one tend to accompany larger values of the other.
Individual observations need not follow the tendency perfectly.
For example, taller trees may tend to have wider trunks.
Positron
Physics 2 · Definition
A positron is the electron’s antiparticle.
It has the same rest mass as an electron but carries positive elementary charge.
For example, beta-plus decay emits a positron along with an electron neutrino.
Potential difference
Physics 2 · Definition
Potential difference is the change in electric potential between two locations.
It represents electric potential-energy change per unit charge and is measured in volts.
For example, the potential difference across a resistor describes the energy transferred per coulomb passing through it.
Potential energy
Physics 1 · Physics 2 · Definition
Potential energy describes energy associated with the configuration of interacting parts of a system.
Its numerical zero is a chosen reference, while differences in potential energy determine physical energy changes.
For example, separating Earth and a ball changes their gravitational potential energy even though the energy belongs to their interaction.
Power
Physics 1 · Physics 2 · Definition
Power is the rate at which energy is transferred or converted.
It is measured in watts, with one watt equal to one joule per second.
For example, two motors can lift the same load through the same height but have different average powers if they take different times.
Power model
Statistics · Definition
A power model has the form y = a times x raised to a constant power b.
For positive values, taking logarithms of both variables gives a linear relationship between log y and log x.
For example, the area of a circle is proportional to the square of its radius.
Power rule
Calculus AB · Calculus BC · Definition
The derivative power rule gives d(xⁿ)/dx=n xⁿ⁻¹ wherever the real power function is differentiable.
Constant multiples remain as constant factors.
For example, the derivative of 3x⁴ is 12x³.
A composite power such as (x²+1)³ also requires the chain rule.
Power series
Calculus BC · Definition
A power series is an infinite series of constant multiples of powers of x−c about a center c.
It can converge for some inputs and diverge for others.
For example, 1+x+x²+⋯ represents 1/(1−x) when |x|<1.
Determining the radius and checking the endpoints establishes its interval of convergence.
Pre-equilibrium approximation
Chemistry · Definition
The pre-equilibrium approximation treats a fast reversible mechanism step as close to equilibrium before a slower step proceeds.
Its equilibrium relation can replace an intermediate concentration in the proposed rate law.
For example, a fast association can express an intermediate's concentration in terms of the reactants that form it.
Precipitate
Chemistry · Definition
A precipitate is a solid phase that forms from species in a solution.
Whether it forms depends on the ion concentrations and the relevant equilibrium conditions.
For example, silver ions and chloride ions can produce a silver chloride precipitate.
Pressure
Physics 1 · Physics 2 · Definition
Pressure is the normal force per unit area exerted on a surface.
It is a scalar measured in pascals, with one pascal equal to one newton per square meter.
For example, a gas exerts pressure throughout its volume through many microscopic collisions, even when its bulk motion is zero.
Probability
Statistics · Definition
Probability quantifies how likely an event is under a specified chance model.
It lies between 0 and 1, with larger values representing greater likelihood.
For example, drawing a red card from a well-shuffled standard deck has probability one half.
Probability distribution
Statistics · Definition
A probability distribution describes a random variable’s possible values and their probabilities.
A discrete distribution lists probability masses, while a continuous distribution describes probabilities through areas under a density.
For example, a distribution can list the probabilities of 0, 1, 2, or 3 red cards in three draws.
Probability model
Statistics · Definition
A probability model specifies possible outcomes and probabilities for events formed from those outcomes.
Probabilities are nonnegative, and the probability of the whole sample space is 1.
In a discrete model, individual outcome probabilities sum to 1; in a continuous density model, total area is 1 even though each single value has probability 0.
For example, a fair four-sided die can be modeled with four outcomes each assigned probability one quarter.
Product rule
Calculus AB · Calculus BC · Definition
The product rule differentiates a product of two differentiable functions.
It gives (fg)′=f′g+fg′, changing one factor at a time and adding the results.
For example, the derivative of x sin x is sin x+x cos x.
The product of the two derivatives is generally not the derivative of the product.
Projectile motion
Physics 1 · Physics 2 · Definition
Projectile motion describes an object after launch when gravity is the only significant force in the chosen model.
Near Earth, horizontal and vertical motion can be analyzed separately using the same time interval and approximately constant downward acceleration.
For example, a ball rolling off a table continues horizontally while its downward velocity increases.
Pure substance
Chemistry · Definition
A pure substance has one chemical identity and a fixed composition.
It may be an element or a compound, so containing more than one element does not make it a mixture.
For example, pure water contains hydrogen and oxygen in a fixed ratio.
Quadrant
Calculus AB · Calculus BC · Definition
The coordinate axes divide the plane into four quadrants.
Quadrant I has two positive coordinates, and the numbering continues counterclockwise.
For example, (−2,3) lies in Quadrant II, where cosine is negative and sine is positive.
Points on an axis are not in any quadrant.
Qualitative analysis
Chemistry · Definition
Qualitative analysis seeks the identities or presence of substances rather than their numerical amounts.
Individual observations often require supporting evidence because different species can produce similar signals.
For example, a supplied flame-color observation can suggest an ion's identity without establishing it uniquely.
Quantitative variable
Statistics · Definition
A quantitative variable records a numerical count or measurement for each individual.
Differences between its values have a meaningful numerical interpretation.
For example, the number of books borrowed and the mass of a backpack are quantitative variables.
Quantization
Physics 2 · Definition
Quantization means a quantity can take only certain allowed values in the stated quantum system.
Bound-state boundary conditions can restrict energy to discrete levels rather than a continuous range.
For example, an electron bound in a hydrogen atom cannot have an arbitrary energy between its allowed levels.
Quartile
Statistics · Definition
Quartiles divide ordered quantitative data into four portions.
The first and third quartiles mark approximately the 25th and 75th percentiles, and sample calculation conventions can differ.
For example, a first quartile of 12 minutes means that roughly one quarter of the recorded times are at or below 12 minutes.
Quotient rule
Calculus AB · Calculus BC · Definition
The quotient rule differentiates a quotient where the denominator is nonzero and both functions are differentiable.
It gives (f/g)′=(f′g−fg′)/g².
For example, differentiating x/(x+1) produces 1/(x+1)² for x≠−1.
The order of the subtraction and the denominator square both matter.
Radian
Calculus AB · Calculus BC · Definition
A radian measures an angle by dividing the intercepted arc length by the circle’s radius.
One full turn is 2π radians, so π radians equals 180 degrees.
For example, an arc with length equal to its radius subtends an angle of one radian.
The familiar derivative formulas for sine and cosine use radian inputs.
Radian
Physics 1 · Physics 2 · Definition
A radian is an angle measure defined by arc length divided by radius.
One complete turn is two pi radians, and angle formulas connecting linear and rotational motion use this measure.
For example, an arc whose length equals the circle’s radius subtends one radian at the center.
Radioactive decay
Physics 2 · Definition
Radioactive decay is a spontaneous nuclear transformation to different nuclei or to a lower energy state of the same nucleus.
Individual decay times are unpredictable, while probabilities describe the behavior of a large sample.
For example, gamma emission can lower a nucleus’s energy without changing its isotope.
Radius of convergence
Calculus BC · Definition
The radius of convergence R describes how far a power series converges from its center.
It converges absolutely for |x−c|<R and diverges for |x−c|>R.
For example, the series 1+x+x²+⋯ has center 0 and radius 1.
When R is finite and positive, convergence at each endpoint must be checked separately.
Random assignment
Statistics · Definition
Random assignment uses chance to allocate experimental units to treatment conditions.
It tends to balance other influences across groups and supports causal inference in a well-conducted experiment, without guaranteeing identical groups.
For example, shuffled labels can assign plants to two fertilizer treatments.
Random sampling
Statistics · Definition
Random sampling uses a specified chance process to select individuals from a population.
It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias.
For example, a random-number generator can select student identifiers from an enrollment list.
Random variable
Statistics · Definition
A random variable assigns a numerical value to each outcome of a chance process.
Its value is uncertain before the outcome is observed, and its distribution gives the probabilities of possible values.
For example, the number of red cards in three draws is a random variable.
Randomized block design
Statistics · Definition
A randomized block design first groups similar experimental units using a relevant characteristic.
Treatments are then randomly assigned within each block, and comparisons account for the blocking.
For example, fertilizer treatments can be randomized separately within sunny and shaded locations.
Range of a function
Calculus AB · Calculus BC · Definition
The range of a function is the set of outputs it actually produces over its domain.
It can differ from the set of outputs that were merely allowed when defining the function.
For example, f(x)=x² on all real inputs has range [0,∞).
Restricting the domain to [1,2] changes that range to [1,4].
Range of data
Statistics · Definition
The range of a data set is its maximum value minus its minimum value.
It measures spread and is sensitive to extreme observations.
For example, values from 3 to 11 have a range of 8.
Rate constant
Chemistry · Definition
A rate constant is the proportionality factor in a specified rate law.
Its value depends on conditions such as temperature, and its units depend on the law's overall order.
For example, a first-order rate constant has units of inverse time when concentration is measured per unit volume.
Rate law
Chemistry · Definition
A rate law relates a reaction rate to reactant concentrations and a rate constant under stated conditions.
For an overall reaction, its concentration exponents are established from evidence rather than copied from the balanced equation.
For example, doubling one reactant concentration doubles the rate when the reaction is first order in that reactant and other conditions are fixed.
Ratio test
Calculus BC · Definition
The ratio test examines the limiting absolute ratio of consecutive series terms.
A limit below 1 gives absolute convergence, and a limit above 1 or infinite gives divergence.
For example, terms 1/n! give ratio 1/(n+1), whose limit is 0, so the series converges absolutely.
A limit equal to 1 is inconclusive.
Rational function
Calculus AB · Calculus BC · Definition
A rational function is a quotient of two polynomials.
Its domain excludes inputs that make the original denominator zero.
For example, (x²−1)/(x−1) agrees with x+1 away from x=1, but remains undefined there.
Canceling a common factor does not automatically restore an excluded input.
RC circuit
Physics 2 · Definition
An RC circuit contains resistance and capacitance that produce time-dependent charging or discharging.
Its initial and long-time behavior depends on the capacitor’s initial voltage and its actual connection to the source and resistors.
For example, a discharged capacitor in a simple series charging loop initially carries current that decreases toward zero as its voltage rises.
RC time constant
Physics 2 · Definition
An RC time constant sets the characteristic time scale of capacitor charging or discharging in a specified resistance–capacitance connection.
For a simple series RC loop it equals resistance times capacitance and is measured in seconds.
For example, increasing the relevant resistance stretches the charging curve when the capacitance remains fixed.
Reaction half-life
Chemistry · Definition
A reaction half-life is the time needed for a reactant's concentration to fall to half its starting value under the specified model.
A first-order reaction has a half-life independent of its starting concentration, while other orders generally do not.
For example, consecutive equal time intervals halve a first-order reactant's concentration repeatedly.
Reaction intermediate
Chemistry · Definition
A reaction intermediate is produced in one mechanism step and consumed in another.
It cancels from the overall equation and differs from a catalyst, which is present before it is consumed and then regenerated.
For example, a short-lived species formed in step one and used in step two can be an intermediate.
Reaction mechanism
Chemistry · Definition
A reaction mechanism proposes a sequence of elementary steps that adds to the net reaction.
It must also be consistent with observed kinetic evidence, though agreement does not prove that the proposal is unique.
For example, an intermediate can form in one step and be consumed in a later step.
Reaction order
Chemistry · Definition
Reaction order describes the concentration dependence in a rate law.
An order in one reactant is its exponent, and overall order is the sum of the concentration exponents.
For example, a second-order dependence on one reactant makes quadrupling the rate follow from doubling that reactant's concentration under fixed conditions.
Reaction quotient
Chemistry · Definition
The reaction quotient describes a mixture's current composition using the equilibrium-expression form.
It uses species activities, approximated by concentrations or partial pressures in suitable models, with powers from balanced coefficients.
For example, a quotient below the equilibrium constant favors net forward reaction toward equilibrium.
Reaction rate
Chemistry · Definition
Reaction rate describes how quickly a reactant or product quantity changes with time.
Rates expressed using different species must account for their stoichiometric coefficients.
For example, a product's concentration–time slope can be used to determine its rate of formation.
Real image
Physics 2 · Definition
A real image forms where light rays physically converge after leaving an optical element.
It can be projected onto a screen placed at the image location.
For example, a projector’s lens forms a real image of its source on a screen.
Real-gas behavior
Chemistry · Definition
Real-gas behavior includes effects of particle attractions and particle volume that the ideal gas law neglects.
Those effects compete, so actual pressure can fall below or rise above the ideal prediction under different conditions.
For example, attraction can lower pressure while excluded particle volume can raise it under strong compression.
Reciprocal derivative rule
Calculus AB · Calculus BC · Definition
The reciprocal derivative rule differentiates 1/f(x) wherever f is differentiable and nonzero.
It gives −f′(x)/[f(x)]².
For example, the derivative of 1/(x²+1) is −2x/(x²+1)².
This is a special case of the quotient rule and can also be derived with the chain rule.
Recursive sequence
Calculus BC · Definition
A recursive sequence specifies later terms using earlier terms and suitable starting values.
The recursion alone is incomplete until those starting values are given.
For example, the Fibonacci sequence can start with 0 and 1 and add the previous two terms to obtain each next term.
To find a later term from the recursion, compute or otherwise determine the earlier terms it depends on.
Redox reaction
Chemistry · Definition
A redox reaction combines oxidation of one species with reduction of another.
Electron loss and gain must balance in the net equation along with atoms and total charge.
For example, zinc reducing copper ions involves zinc oxidation and copper-ion reduction.
Reducing agent
Chemistry · Definition
A reducing agent causes another species to gain electrons.
The agent supplies those electrons and is itself oxidized.
For example, zinc acts as a reducing agent when it transfers electrons to copper ions.
Reduction
Chemistry · Definition
Reduction is electron gain, represented by a decrease in oxidation number.
It is paired with oxidation in an electron-transfer reaction.
For example, a copper ion with charge +2 gaining two electrons to form copper metal undergoes reduction.
Reference frame
Physics 1 · Physics 2 · Definition
A reference frame specifies the observer and coordinates used to describe positions and motion.
Different observers can assign different velocities to the same object.
For example, a seated passenger is at rest relative to a train but moving relative to the station.
Reflection
Physics 2 · Definition
Reflection occurs when a wave returns from a boundary into the region from which it arrived.
For a light ray at a surface, the reflection angle equals the incidence angle when both are measured from the local normal.
For example, a smooth mirror produces an orderly image while a rough surface scatters reflected rays in many directions.
Refraction
Physics 2 · Definition
Refraction is a change in wave propagation at a boundary where wave speed changes.
For light entering a new medium, frequency stays fixed while speed and wavelength can change, and the direction generally bends unless incidence is normal.
For example, a light ray entering water obliquely from air bends toward the normal.
Refractive index
Physics 2 · Definition
Refractive index is the ratio of light’s speed in vacuum to its speed in the material at the relevant frequency.
It is dimensionless and can depend on wavelength.
For example, glass usually has a larger refractive index than air, so visible light travels more slowly through the glass.
Regression intercept
Statistics · Definition
A regression intercept is the line’s predicted response when the explanatory variable is zero.
It may lack a useful contextual interpretation when zero lies outside the relevant data range.
For example, an intercept predicts height at age zero even if measurements were collected only for mature plants.
Regression slope
Statistics · Definition
A regression slope is the predicted change in the response for a one-unit increase in the explanatory variable.
Its units are response units per explanatory-variable unit.
For example, a slope of 2 centimeters per week predicts 2 additional centimeters of height for each additional week.
Relative frequency
Statistics · Definition
Relative frequency is a group’s count divided by the total count.
It can be written as a proportion or multiplied by 100 to give a percentage.
For example, 8 choices out of 20 have relative frequency 0.4, or 40%.
Relative isotope abundance
Chemistry · Definition
Relative isotope abundance is the fraction or percentage of an element's atoms belonging to a particular isotope.
It concerns the number of atoms rather than their fraction of the sample's mass.
For example, an isotope present in 25 percent of the atoms contributes a factor of 0.25 to the weighted average atomic mass.
Removable discontinuity
Calculus AB · Calculus BC · Definition
A removable discontinuity occurs when a finite two-sided limit exists but the function is missing or has a different value at the target.
Changing or defining that single value to equal the limit repairs continuity.
For example, (x²−1)/(x−1) has a removable discontinuity at 1 and can be repaired by setting its value there to 2.
A jump with unequal one-sided limits cannot be repaired this way.
Replication
Statistics · Definition
Replication uses multiple independent experimental units for each treatment.
It helps estimate chance variation and increases the information available for comparing treatment responses.
For example, assigning many independently treated plants to each fertilizer is replication, while repeatedly measuring one plant adds a different kind of information.
Residual
Statistics · Definition
A residual is an observed response minus its predicted response.
A positive residual places the observed point above the fitted line, while a negative residual places it below.
For example, an observed height of 18 centimeters and a predicted height of 16 centimeters give a residual of 2 centimeters.
Residual plot
Statistics · Definition
A residual plot displays prediction residuals against explanatory values or fitted responses.
Curvature can suggest that a straight-line model is inappropriate, while changing spread can challenge constant-variance assumptions.
For example, a U-shaped residual pattern suggests that the line misses a curved relationship.
Resistant measure
Statistics · Definition
A resistant measure changes relatively little when a small number of observations become extreme.
Resistance means reduced sensitivity, rather than complete immunity to every change in the data.
For example, changing a largest value from 30 to 300 may leave the median unchanged while substantially increasing the mean.
Resistivity
Physics 2 · Definition
Resistivity is a material property describing resistance to electrical current under specified conditions.
It is distinct from the resistance of a particular object and is measured in ohm-meters.
For example, changing a wire’s length changes its resistance without changing the material’s resistivity when temperature and composition stay fixed.
Resonance
Physics 2 · Definition
Resonance is a large oscillatory response when a periodic driving influence has a frequency near a system’s natural frequency.
The response also depends on damping and on how the driving influence couples to the system.
For example, repeated pushes timed to a swing’s motion can build its amplitude.
Resonance in bonding
Chemistry · Definition
Resonance uses multiple Lewis contributors with the same atom positions to describe delocalized electrons.
The actual species has one electronic structure rather than switching back and forth between the drawings.
For example, nitrate's equivalent N–O bonds are described with several resonance contributors.
Response bias
Statistics · Definition
Response bias is a systematic pattern of inaccurate or influenced answers.
Wording, interviewer behavior, memory, and social pressure can affect what people report.
For example, a leading question can encourage respondents to favor a proposed school policy.
Response variable
Statistics · Definition
A response variable records the outcome being studied.
Its relationship to an explanatory variable can be investigated without assuming that the explanatory variable causes it.
For example, plant height is a response when researchers compare fertilizer treatments.
Restoring force
Physics 1 · Physics 2 · Definition
A restoring force points toward an equilibrium position when an object is displaced from it.
Simple harmonic motion requires the net restoring force to be proportional to displacement and opposite in direction.
For example, a compressed ideal spring pushes its attached object back toward the spring’s relaxed length.
Resultant vector
Physics 1 · Physics 2 · Definition
A resultant vector is the vector sum of two or more vectors.
Its components are the sums of the corresponding components of the vectors being combined.
For example, two equal forces in opposite directions have a zero resultant even though both forces are present.
Reversible reaction
Chemistry · Definition
A reversible reaction can proceed in both forward and reverse directions under the modeled conditions.
Its net direction depends on the current composition and temperature.
For example, a product-rich starting mixture can react backward toward equilibrium.
Riemann rearrangement theorem
Calculus BC · Definition
The Riemann rearrangement theorem concerns changing the order of terms in a conditionally convergent real series.
Such a series can be rearranged to converge to any chosen real value or to diverge.
For example, the alternating harmonic series has rearrangements with a sum different from its original one.
An absolutely convergent series instead retains its sum under every rearrangement.
Riemann sum
Calculus AB · Calculus BC · Definition
A Riemann sum approximates accumulation by adding sample heights times subinterval widths.
Left, right, and midpoint sums choose different sample positions in each subinterval.
For example, a left sum for x² on [0,2] with two equal widths uses heights 0 and 1, giving 1.
A definite integral is the limiting value as the largest subinterval width approaches zero when that limit exists.
Right-hand rule
Physics 1 · Physics 2 · Definition
A right-hand rule is a convention for determining the direction of a vector cross product or a current-produced magnetic field.
The chosen gesture must use the correct input vectors in their correct order, with reversal when a negative charge requires it.
For example, the thumb-and-curl rule around a straight wire uses conventional current as the thumb direction.
Right-skewed distribution
Statistics · Definition
A right-skewed distribution has a longer tail toward larger values.
Most observations may lie toward the lower end while a few large values extend the right tail.
For example, waiting times can be right-skewed when most waits are short and a few are very long.
Rolle’s theorem
Calculus AB · Calculus BC · Definition
Rolle’s theorem is the equal-endpoint case of the Mean Value Theorem.
A function continuous on [a,b], differentiable on (a,b), and satisfying f(a)=f(b) has an interior point with derivative 0.
For example, x² on [−1,1] has equal endpoint values and derivative 0 at x=0.
A sharp corner can break the differentiability hypothesis even when the endpoints agree.
Rolling without slipping
Physics 1 · Physics 2 · Definition
Rolling without slipping means the point of contact has no instantaneous relative motion along the contacting surface.
For a wheel rolling on a stationary surface, the center-of-mass speed equals angular speed times radius.
For example, a wheel rolling smoothly along a road has a momentarily stationary contact point while its top moves faster than its center.
Root test
Calculus BC · Definition
The root test examines the limiting nth root of the absolute value of the nth series term.
A limit below 1 gives absolute convergence, and a limit above 1 or infinite gives divergence.
For example, terms (1/3)ⁿ have nth-root limit 1/3 and their series converges absolutely.
A limit equal to 1 is inconclusive.
Root-mean-square speed
Physics 2 · Definition
Root-mean-square speed is the square root of the mean of the particles’ squared speeds.
It represents a kinetic-energy average and is different from both average speed and vector-average velocity.
For example, an equilibrium gas can have zero vector-average velocity while its rms speed is substantial.
Rotational equilibrium
Physics 1 · Physics 2 · Definition
Rotational equilibrium means the net external torque about the chosen axis is zero.
A rigid body’s angular velocity then stays constant when its rotational inertia is fixed; it need not be at rest.
For example, opposing torques can balance on a wheel that is already rotating steadily.
Rotational inertia
Physics 1 · Physics 2 · Definition
Rotational inertia measures how strongly a mass distribution resists a change in angular velocity about a specified axis.
Mass farther from that axis contributes more strongly because each contribution depends on squared distance.
For example, moving small masses outward along a rotating rod increases its rotational inertia without changing its total mass.
Rotational kinetic energy
Physics 1 · Physics 2 · Definition
Rotational kinetic energy is the kinetic energy associated with rotation about an axis.
For a rigid body about a fixed axis it is one half of rotational inertia times angular speed squared.
For example, a rolling wheel can possess both translational kinetic energy of its center of mass and rotational kinetic energy about that center.
Salt bridge
Chemistry · Definition
A salt bridge provides an ionic conducting path between separated half-cells.
Ion movement helps prevent bulk charge buildup while electrons move through the external circuit.
For example, bridge anions can move toward a half-cell where metal cations are being produced.
Sample
Statistics · Definition
A sample is the subset of a population actually studied.
How it is selected matters for the conclusions that can be drawn about the population.
For example, 60 selected students may be a sample from a school’s full enrollment.
Sample proportion
Statistics · Definition
A sample proportion is the number of sampled individuals with a specified characteristic divided by the sample size.
It is commonly written p-hat and estimates the corresponding population proportion.
For example, 18 walkers among 60 sampled students give a sample proportion of 0.30.
Sample space
Statistics · Definition
The sample space is the set of all possible outcomes of a chance process.
Outcomes must be described at the level relevant to the question being modeled.
For example, the sample space for one draw from a red-blue bag can be red and blue.
Sample statistic
Statistics · Definition
A statistic is a numerical summary calculated from sample data.
Its value can change when a different sample is selected from the same population.
For example, the proportion of walkers among 60 sampled students is a statistic.
Sampling distribution
Statistics · Definition
A sampling distribution describes how a statistic varies across all possible samples drawn under a specified sampling design and sample size.
It is a distribution of statistics rather than of individual observations.
For example, the proportions from repeated random samples of 60 students form a sampling distribution.
Sampling distribution of a sample mean
Statistics · Definition
The sampling distribution of a sample mean describes averages from repeated samples of a fixed size.
For independent identically distributed observations with finite variance, its mean is the population mean and its standard deviation is the population standard deviation divided by the square root of the sample size.
For example, averages from samples of 100 measurements vary less than averages from samples of 25.
Sampling distribution of a sample proportion
Statistics · Definition
The sampling distribution of a sample proportion describes proportions obtained from repeated samples of a fixed size.
A Normal approximation requires an appropriate design and sufficiently large expected success and failure counts.
For example, larger independent samples usually give sample proportions that vary less around the population proportion.
Sampling error
Statistics · Definition
Sampling error is the difference arising because a sample rather than the entire population is observed.
Random sampling variability is distinct from systematic coverage or response bias.
For example, two properly selected random samples can produce different mean commuting times purely by chance.
Sampling frame
Statistics · Definition
A sampling frame is the list or other operational source from which a sample is selected.
It may differ from the target population if some members are omitted or listed incorrectly.
For example, a current enrollment list can serve as a frame for selecting students.
Sampling variability
Statistics · Definition
Sampling variability describes how an estimate changes across repeated samples drawn with the same design.
Larger suitable random samples generally reduce this variability, without automatically correcting systematic bias.
For example, means from samples of 100 students usually vary less than means from samples of 10 students.
Sampling with replacement
Statistics · Definition
Sampling with replacement returns a selected item to the pool before the next selection.
With independent random selections, the available composition and selection probabilities remain unchanged.
For example, replacing a drawn card before reshuffling allows that same card to be drawn again.
Sampling without replacement
Statistics · Definition
Sampling without replacement does not return selected items before later selections.
Selections from a finite population are dependent, although treating them as approximately independent can be reasonable when the sample is a small fraction of the population.
For example, selecting students without replacement ensures that no student is chosen twice.
Saturated solution
Chemistry · Definition
A saturated solution has the dissolved concentration corresponding to equilibrium with the undissolved phase under the stated conditions.
Microscopic dissolution and precipitation can continue at equal rates when the solid is present.
For example, additional solid may remain undissolved after a saturated salt solution is reached at fixed temperature.
Scalar
Physics 1 · Physics 2 · Definition
A scalar is a quantity described by a value and unit without a spatial direction.
A scalar can have a sign when its definition or reference permits one; scalar does not mean always positive.
For example, mass is a scalar, whereas the velocity of that mass also needs a direction.
Scatterplot
Statistics · Definition
A scatterplot displays paired values of two quantitative variables as points.
The horizontal coordinate represents one variable and the vertical coordinate represents the other.
For example, plotting hours studied against test scores helps reveal a pattern without establishing causation.
Secant line
Calculus AB · Calculus BC · Definition
A secant line passes through two distinct points on a curve.
Its slope is the average rate of change between their inputs.
For example, the secant to y=x² through inputs 1 and 3 has slope (9−1)/(3−1)=4.
Tangent slopes arise from limits of these secant slopes when the second point approaches the first.
Second derivative
Calculus AB · Calculus BC · Definition
A second derivative is the derivative of the first derivative.
It measures how the first rate of change changes with the input; further repeated differentiation gives higher-order derivatives.
For example, f(x)=x³ has f′(x)=3x² and f″(x)=6x.
For position measured against time, the second derivative is acceleration.
Second derivative test
Calculus AB · Calculus BC · Definition
The second derivative test classifies a stationary point when its second derivative is nonzero.
If f′(c)=0, a positive f″(c) gives a local minimum and a negative f″(c) gives a local maximum under the theorem’s smoothness conditions.
For example, f(x)=x² has f′(0)=0 and f″(0)=2, so 0 is a local minimum.
If f″(c)=0, the test is inconclusive and another argument is required.
Second law of thermodynamics
Physics 2 · Definition
The second law of thermodynamics states that the total entropy of an isolated system cannot decrease.
It explains the preferred direction of spontaneous processes without replacing conservation of energy.
For example, a cooler object does not spontaneously transfer net thermal energy to a hotter object without another compensating process.
Separation of variables
Calculus AB · Calculus BC · Definition
Separation of variables solves certain differential equations by placing the dependent and independent variables on opposite sides.
For y′=g(x)h(y), rewrite dy/h(y)=g(x) dx where h(y)≠0 and integrate.
For example, y′=xy leads to ln|y|=x²/2+C for nonzero solutions.
Check any solutions lost by division, such as y=0 in this example.
Sequence
Calculus BC · Definition
A sequence is an ordered list of terms indexed by integers.
A rule can specify each term directly or recursively from earlier terms.
For example, aₙ=1/n produces 1, 1/2, 1/3, and so on.
An infinite sequence converges when its terms approach a finite limit as the index grows without bound.
Series connection
Physics 2 · Definition
A series connection places elements along one unbranched path.
The same current passes through the series elements, while their potential differences can differ.
For example, adding another resistor to a simple series loop raises the total resistance and reduces current from a fixed ideal source.
Short circuit
Physics 2 · Definition
A short circuit provides a very low-resistance path between nodes that would otherwise have a potential difference.
It can bypass components and draw a large current limited by the real source and wiring resistance.
For example, a wire directly across an ideal resistor can bypass that resistor; a real battery short can heat the wiring dangerously.
Sigma bond
Chemistry · Definition
A sigma bond has bonding electron density along the line joining two nuclei.
It can arise from end-to-end overlap in a localized orbital model.
For example, the carbon–carbon bond in ethane is a sigma bond.
Significance level
Statistics · Definition
A significance level is a threshold selected before testing to limit Type I errors under the null model.
A p-value at or below that threshold leads to rejection under the stated rule, while a larger p-value leads to failure to reject.
For example, an alpha level of 0.01 demands stronger incompatibility with the null model than a level of 0.05.
Significance test
Statistics · Definition
A significance test assesses how compatible observed data are with a specified null model.
It compares a test statistic or p-value with a decision rule chosen for the study.
For example, a test can assess whether a sample provides evidence that a population walking proportion differs from a stated value.
Simple harmonic motion
Physics 1 · Physics 2 · Definition
Simple harmonic motion occurs when the net restoring force is proportional to displacement from equilibrium and points in the opposite direction.
The resulting position varies sinusoidally with time in the ideal model.
For example, a mass attached to an ideal spring can execute simple harmonic motion when damping is negligible.
Simple random sample
Statistics · Definition
A simple random sample of a fixed size gives every possible subset of that size the same chance of selection.
Giving individuals equal selection chances alone does not establish that the design is an SRS.
For example, randomly choosing three identifiers without replacement from ten gives an SRS when every three-identifier set is equally likely.
Simulation
Statistics · Definition
A simulation imitates a chance process using a specified model.
Repeated simulated outcomes can approximate probabilities when the model faithfully represents the situation.
For example, random digits can model repeated draws from a bag with known color proportions.
Sinusoidal function
Calculus AB · Calculus BC · Definition
A sinusoidal function has the shape of a sine or cosine graph after shifts, scaling, and possible reflection.
A common form is A sin(B(x−h))+D with nonzero A and B.
For example, 2 sin(3x)+1 has amplitude 2, midline 1, and period 2π/3.
The parameters control vertical variation, repetition, and position of the wave.
Skewness
Statistics · Definition
Skewness describes asymmetry in a quantitative distribution’s tails.
A longer right tail indicates right skew, while a longer left tail indicates left skew.
For example, many short waits with a few very long waits produce right skew.
Slope field
Calculus AB · Calculus BC · Definition
A slope field displays short line segments with slopes specified by a first-order differential equation.
For y′=F(x,y), the segment at (x,y) has slope F(x,y).
For example, y′=y gives horizontal segments along y=0 and positive slopes above that axis.
Solution curves follow the local slopes, and an initial point helps choose the relevant curve.
Snell’s law
Physics 2 · Definition
Snell’s law relates the incidence and refraction angles to the refractive indices of two media.
Both angles are measured from the normal to the boundary, rather than from the surface.
For example, entering a higher-index material at oblique incidence bends a light ray toward the normal.
Solid of revolution
Calculus AB · Calculus BC · Definition
A solid of revolution is formed by rotating a plane region around a specified line.
The axis and region determine the resulting radii and any central hole.
For example, rotating the region under y=x on [0,1] around the horizontal axis produces a cone.
Disc, washer, and shell methods describe different ways to slice or assemble such a solid.
Solubility
Chemistry · Definition
Solubility describes how much of a substance can dissolve in a specified solvent under stated conditions.
It depends on the solute, solvent, temperature, and other relevant conditions and is distinct from how fast dissolution occurs.
For example, two salts can have different equilibrium solubilities in water at the same temperature.
Solubility product
Chemistry · Definition
A solubility product is an equilibrium constant for a specified dissolution reaction.
Its expression includes dissolved-species activities and omits the pure solid.
For example, dissolving one formula unit that releases two anions gives an anion activity squared in the expression.
Solute
Chemistry · Definition
A solute is a component dispersed in a solvent to form a solution.
It need not be a solid, and a solution need not be liquid.
For example, carbon dioxide is a gaseous solute in carbonated water.
Solvation
Chemistry · Definition
Solvation is the interaction of solvent particles with dispersed solute particles.
When the solvent is water, the process is called hydration.
For example, oriented water molecules form hydration shells around dissolved ions.
Solvent
Chemistry · Definition
A solvent is the component used as the dissolving medium in a solution.
It is often present in the greater amount, though the designation also depends on the mixture's context.
For example, water is the solvent in a dilute aqueous sugar solution.
Specific gravity
Physics 2 · Definition
Specific gravity is a material’s density divided by a chosen reference density.
It is dimensionless; for liquids and solids the reference is usually water at a specified temperature.
For example, a liquid with specific gravity below one is less dense than the reference water.
Specific heat
Physics 2 · Definition
Specific heat is the energy needed per unit mass for a unit temperature increase under the stated conditions.
Its units are joules per kilogram per kelvin, and the simple calorimetry model treats it as constant without a phase change.
For example, equal masses of two materials can require different energy transfers for the same temperature rise.
Specific heat capacity
Chemistry · Definition
Specific heat capacity is the heat required per unit mass for a one-unit temperature increase under stated conditions.
It connects a mass and measured temperature change to heat transfer when the model applies.
For example, water requires about 4.18 joules per gram for a one-kelvin temperature rise near room temperature.
Spectator ion
Chemistry · Definition
A spectator ion appears unchanged on both sides of the complete ionic equation.
It is omitted from the net ionic equation because it does not undergo the represented reaction.
For example, sodium ions can be spectators when two other dissolved ions form a precipitate.
Spectroscopy
Chemistry · Definition
Spectroscopy studies interactions between matter and electromagnetic radiation.
The absorbed or emitted frequencies can reveal particular energy changes in a species.
For example, infrared absorption often corresponds to changes in molecular vibration.
Spring constant
Physics 1 · Physics 2 · Definition
A spring constant measures the stiffness of a spring within its approximately linear range.
Its units are newtons per meter, and a larger value means more force is needed for the same deformation.
For example, two springs stretched by the same distance exert different forces if one is stiffer.
Spring potential energy
Physics 1 · Physics 2 · Definition
Spring potential energy is energy stored in an ideal spring’s deformation from its relaxed length.
It is proportional to the square of deformation, so equal extension and compression magnitudes store equal energy.
For example, compressing a spring before releasing a toy transfers energy that can later become kinetic energy.
Squeeze theorem
Calculus AB · Calculus BC · Definition
The squeeze theorem determines a limit by trapping a function between two others with the same limit.
The bounding inequalities must hold sufficiently near the target where the functions are being compared.
For example, −x²≤x² sin(1/x)≤x² for x≠0 shows that the middle expression approaches 0 as x approaches 0.
A picture suggesting a trap does not replace the required inequality.
Standard deviation
Statistics · Definition
Standard deviation measures spread around the mean using squared deviations and a square root.
A sample standard deviation uses the sum of squared deviations divided by one less than the sample size before taking the square root.
For example, a larger standard deviation of travel times indicates greater variability, in the original time units.
Standard deviation of residuals
Statistics · Definition
The standard deviation of residuals estimates the typical prediction error around a fitted regression line.
For simple linear regression it is the square root of the sum of squared residuals divided by the sample size minus 2.
For example, a residual standard deviation of 3 points indicates scatter around predictions on the scale of a few score points.
Standard enthalpy of formation
Chemistry · Definition
Standard enthalpy of formation is the enthalpy change for forming one mole of a substance from its elements in their standard reference forms.
The substance's phase and specified temperature matter.
For example, an element in its standard reference form has zero standard formation enthalpy by convention.
Standard error
Statistics · Definition
A standard error describes the standard deviation of a statistic’s sampling distribution, usually estimated from data.
It measures uncertainty in an estimate rather than the spread of individual observations.
For example, the estimated standard error of a sample mean is the sample standard deviation divided by the square root of the sample size.
Standard molar entropy
Chemistry · Definition
Standard molar entropy is the molar entropy of a substance in its specified standard state at a stated temperature.
It is an absolute-property value rather than an entropy of formation.
For example, an element can have a positive standard molar entropy even though its standard formation enthalpy is zero.
Standard Normal distribution
Statistics · Definition
The standard Normal distribution has mean 0 and standard deviation 1.
Standardizing a Normal random variable produces this common probability scale.
For example, a standardized value of 2 lies two standard deviations above the standard Normal mean.
Standard state
Chemistry · Definition
A standard state is a defined reference for a substance's thermodynamic properties.
It specifies activities or reference forms and does not by itself fix the temperature at 25 °C.
For example, gases commonly use a one-bar reference pressure while the table states temperature separately.
Standing wave
Physics 2 · Definition
A standing wave is a spatially fixed interference pattern formed by suitable waves traveling in opposite directions.
Its nodes remain at zero displacement while antinodes mark positions of maximum oscillation amplitude.
For example, a string fixed at both ends can support standing waves whose wavelengths fit the boundary conditions.
State variable
Physics 2 · Definition
A state variable is determined by a system’s present equilibrium state rather than by the route used to reach it.
Internal energy and entropy are state properties, whereas heat and work describe transfers along a process.
For example, two gas processes with identical endpoints have the same internal-energy change but can involve different amounts of work.
Static friction
Physics 1 · Physics 2 · Definition
Static friction opposes the tendency for relative sliding between surfaces that are not sliding at their contact.
In the simple model it adjusts up to a maximum set by the coefficient of static friction and normal force.
For example, static friction can push a walking person forward when the foot pushes backward on the ground.
Stationary phase
Chemistry · Definition
The stationary phase is the material that remains in place in a chromatographic system.
Components differ in how strongly they interact with it relative to the mobile phase.
For example, stronger stationary-phase retention can make one dye move less far than another under the same conditions.
Stationary point
Calculus AB · Calculus BC · Definition
A stationary point is a point where a function’s derivative is zero.
Its tangent is horizontal when the derivative exists as a finite number.
For example, x³ has a stationary point at 0 but continues increasing through it.
A stationary point can be a local extremum or neither, so classification requires additional evidence.
Statistical bias
Statistics · Definition
Statistical bias is a systematic tendency to favor some outcomes or miss a target value.
For an estimator, bias is its long-run average estimate minus the parameter it estimates.
For example, a survey that consistently misses evening workers may systematically misrepresent commuting patterns.
Statistical independence
Statistics · Definition
Statistical independence means that learning one outcome does not change the probability distribution of another.
For two events, it implies that the probability of both occurring equals the product of their probabilities.
For example, independent draws with replacement keep the second draw’s color probabilities unchanged by the first.
Statistical inference
Statistics · Definition
Statistical inference uses data and a study design to draw conclusions beyond the observed values.
Random sampling supports population generalization, while well-conducted random assignment supports treatment-effect conclusions.
For example, a random sample can estimate a school-wide preference without surveying every student.
Statistical power
Statistics · Definition
Statistical power is the probability of rejecting the null hypothesis for a specified true alternative.
It equals 1 minus the Type II error probability for that alternative and can increase with a larger sample or a larger true effect.
For example, a study with power 0.8 detects its specified alternative in about 80% of repeated studies under the model.
Editorial qualification: The captured text describes power as detecting a Type II error. Power is the probability of correctly rejecting the null hypothesis for a specified true alternative, equal to 1 − β.
Statistical significance
Statistics · Definition
A result is statistically significant at a stated level when it meets the test’s rejection rule.
Statistical significance alone does not establish a large effect, practical importance, or causation.
For example, a very large sample can detect a small population difference that has little practical importance.
Stefan–Boltzmann constant
Physics 2 · Definition
The Stefan–Boltzmann constant is the proportionality constant in the ideal blackbody law linking emitted power to surface area and the fourth power of kelvin temperature.
Its value is approximately 5.67 × 10⁻⁸ W/(m²·K⁴), so multiplying it by area and temperature to the fourth power gives watts.
For example, the same constant applies to two ideal blackbodies of different sizes, while their areas and temperatures determine their different emitted powers.
Stefan–Boltzmann law
Physics 2 · Definition
The Stefan–Boltzmann law gives an ideal blackbody’s total emitted power as proportional to surface area and kelvin temperature to the fourth power.
Total emitted power differs from net radiative transfer because surrounding objects can also supply radiation.
For example, doubling a blackbody’s surface area doubles its emitted power when its temperature stays fixed.
Stemplot
Statistics · Definition
A stemplot splits each numerical observation into a stem and a leaf.
A key specifies the place values, so the original observations can be recovered.
For example, a key of 3 | 7 = 37 means that stem 3 with leaf 7 represents 37.
Stoichiometry
Chemistry · Definition
Stoichiometry uses a balanced chemical equation to relate amounts of reactants and products.
Equation coefficients provide mole ratios rather than direct mass ratios.
For example, forming water from hydrogen and oxygen consumes two moles of hydrogen molecules per mole of oxygen molecules.
Stopping potential
Physics 2 · Definition
Stopping potential is the smallest retarding-voltage magnitude that stops the fastest photoelectrons from reaching the collector.
Its electric energy barrier measures the maximum kinetic energy of the emitted electrons.
For example, increasing above-threshold light frequency raises the maximum kinetic energy and therefore raises the required stopping-potential magnitude.
Stratified random sampling
Statistics · Definition
Stratified random sampling divides the population into nonoverlapping groups and randomly samples within every group.
The strata can improve representation and precision when their members share a relevant characteristic.
For example, a school may randomly sample students separately within each grade level.
Stratum
Statistics · Definition
A stratum is a nonoverlapping subgroup used in stratified sampling.
A stratified design samples within every stratum rather than selecting only some groups.
For example, each grade level can be a stratum in a school survey.
Streamline
Physics 1 · Physics 2 · Definition
A streamline is a curve tangent to the local fluid velocity at each point at one instant.
In steady flow its shape stays fixed and fluid particles follow it, while in changing flow a streamline need not be a particle’s full path.
For example, streamline arrows around a smooth obstacle show the local direction of the passing fluid.
Strong acid or base
Chemistry · Definition
A strong acid transfers protons essentially completely to water in the applicable aqueous-solution model.
A dissolved strong hydroxide base dissociates into its constituent ions and supplies hydroxide; it does not have to react with water to generate that hydroxide.
Strength is distinct from concentration and, for a solid base, from how much can dissolve.
For example, dissolved sodium hydroxide supplies sodium and hydroxide ions even when the prepared solution is dilute.
Strong interaction
Physics 2 · Definition
The strong interaction binds protons and neutrons at nuclear distances.
It acts over a very short effective range between nucleons, while electric repulsion between protons also affects nuclear stability.
For example, bringing nuclei close enough for fusion requires overcoming their electric repulsion before nuclear binding can occur.
Student’s t-distribution
Statistics · Definition
A Student’s t-distribution is a symmetric reference distribution with heavier tails than the standard Normal distribution.
Its shape depends on degrees of freedom and approaches the standard Normal shape as those degrees of freedom grow.
For example, inference about a Normal population mean with an unknown standard deviation uses a t reference with one fewer degree of freedom than the sample size.
Sublimation
Chemistry · Definition
Sublimation is a phase change directly from solid to gas without an intervening liquid phase.
Whether it occurs depends on the substance and pressure–temperature conditions.
For example, solid carbon dioxide sublimes at ordinary atmospheric pressure.
Substitution for integration
Calculus AB · Calculus BC · Definition
Substitution rewrites an integral using a new variable whose differential accounts for a derivative factor.
It reverses the chain rule for suitable compositions.
For example, with u=x²+1 and du=2x dx, the integral of 2x(x²+1)³ becomes the integral of u³.
For a definite integral, transform the bounds consistently or return to the original variable before evaluating them.
Superposition
Physics 1 · Physics 2 · Definition
Superposition means adding the separate contributions to obtain a total response in a linear model.
Vector fields add vectorially, scalar potentials add algebraically, and wave disturbances add with their signs at each place and time.
For example, equal but opposite electric-field contributions can give zero net field at one point while both sources remain present.
Survey
Statistics · Definition
A survey collects information by asking individuals questions.
Its quality depends on the sample design, question wording, and responses received.
For example, a school can survey selected families about their transportation needs.
Symmetric distribution
Statistics · Definition
A symmetric distribution has approximately matching shapes on either side of its center.
Symmetry alone does not establish that a distribution is Normal.
For example, a uniform distribution over a centered interval is symmetric but has a flat rather than bell-shaped density.
Systematic random sampling
Statistics · Definition
Systematic random sampling chooses a random starting point and then selects at a fixed interval in an ordered frame.
A repeating pattern in the frame can make the interval an unsuitable choice.
For example, after a random start among the first ten entries, a researcher can select every tenth entry.
T-table
Statistics · Definition
A t-table lists critical values for t-distributions at specified degrees of freedom and tail probabilities.
Its column convention must be checked before using it for a one-sided test or a two-sided interval.
For example, a two-sided 95% interval leaves 2.5% in each tail when selecting its critical value.
Tangent line
Calculus AB · Calculus BC · Definition
A tangent line at a differentiable point passes through the point and has slope equal to the derivative there.
It describes the function’s local linear behavior.
For example, y=x² has tangent line y=4+4(x−2) at x=2.
A tangent can cross the curve, so touching at exactly one point is not its general definition.
Taylor polynomial
Calculus BC · Definition
A Taylor polynomial approximates a function near a center using its value and successive derivatives at that center.
The degree-n polynomial uses the terms through the nth power of x−c, with derivative coefficients divided by factorials.
For example, a degree-2 Taylor polynomial for eˣ at 0 is 1+x+x²/2.
An error argument is needed to judge the approximation away from the center.
Taylor series
Calculus BC · Definition
A Taylor series is the infinite power series built from a function’s derivatives at a chosen center.
Its coefficients are the derivatives at the center divided by the corresponding factorials.
For example, eˣ has Taylor series 1+x+x²/2!+x³/3!+⋯ centered at 0.
A Taylor series represents the function only where convergence and the vanishing remainder justify that equality.
Telescoping series
Calculus BC · Definition
A telescoping series has partial sums in which cancellation reduces the result to a fixed number of boundary terms.
Write out a partial sum before taking its limit so that the cancellation is justified.
For example, the first n terms of 1/[k(k+1)]=1/k−1/(k+1) sum to 1−1/(n+1).
Taking that limit shows the series starting at k=1 sums to 1.
Temperature
Physics 2 · Definition
Temperature is a state quantity that determines the direction of net energy transfer through thermal contact.
For a classical ideal gas it is proportional to the average translational kinetic energy per particle, rather than to the total energy of the sample.
For example, a small hot metal sample can have a higher temperature but less total internal energy than a large cooler sample.
Tension
Physics 1 · Physics 2 · Definition
Tension is a pulling force transmitted through a stretched rope, string, or similar connector.
It acts along the connector away from the object it pulls, and an ideal massless rope has the same tension throughout an uninterrupted segment.
For example, a rope supporting an accelerating load can have a tension different from the load’s weight.
Terminal speed
Physics 2 · Definition
Terminal speed is a constant speed reached when the forces along the direction of motion balance.
For a falling object in fluid, increasing drag can eventually balance weight after buoyancy is also accounted for.
For example, a falling raindrop can reach approximately constant downward velocity instead of accelerating indefinitely.
Test statistic
Statistics · Definition
A test statistic summarizes how the sample departs from the null model.
Its reference distribution under the null supplies the probabilities used in testing.
For example, a z-statistic expresses a sample proportion’s difference from a hypothesized proportion in null-model standard-error units.
Thermal conduction
Physics 2 · Definition
Thermal conduction transfers energy through microscopic interactions without bulk flow of the material.
Particles and, in metals, mobile electrons can carry energy from hotter regions toward cooler regions.
For example, the handle of a metal spoon warms when its other end is placed in hot water.
Thermal conductivity
Physics 2 · Definition
Thermal conductivity measures how readily a material conducts energy through a temperature gradient.
It is a material property with units watts per meter per kelvin, distinct from the sample’s heat capacity or thickness.
For example, a metal and an insulating foam slab of the same dimensions can have very different conduction rates.
Thermal equilibrium
Chemistry · Definition
Thermal equilibrium means that bodies in thermal contact have the same temperature and no net heat transfer between them.
It does not mean that they contain equal amounts of energy or have the same mass.
For example, a small metal sample and a large water bath can reach the same temperature.
Thermal equilibrium
Physics 2 · Definition
Thermal equilibrium between systems means their temperatures are equal and there is no net energy transfer between them through thermal processes.
Microscopic energy exchanges can continue while their opposing contributions balance.
For example, water and a metal sample left together in an insulated cup approach a common equilibrium temperature.
Thermal expansion
Physics 2 · Definition
Thermal expansion is a change in a material’s dimensions as its temperature changes.
A simple coefficient model approximates the change over a limited temperature range and depends on the material and geometry.
For example, leaving gaps between sections of a metal rail allows space for expansion in warmer conditions.
Thermal radiation
Physics 2 · Definition
Thermal radiation is electromagnetic energy emitted by matter because of its temperature.
It can transfer energy through a vacuum, and net radiative transfer between ordinary bodies is from hotter to cooler.
For example, sunlight warming a surface does not need air to carry the energy across space.
Thermodynamic favorability
Chemistry · Definition
Thermodynamic favorability describes whether a process has a driving force in the stated direction under specified conditions.
It does not determine how quickly the process occurs.
For example, a reaction with a favorable free-energy change can persist without visible progress when a large kinetic barrier blocks it.
Thin lens
Physics 2 · Definition
A thin-lens model treats a lens’s thickness as negligible compared with the relevant object and image distances.
The model relates focal length to signed object and image distances and is most accurate for paraxial rays.
For example, a converging lens can form a real or a virtual image depending on the object’s position relative to its focal point.
Thin-film interference
Physics 2 · Definition
Thin-film interference occurs when waves reflected from different film interfaces overlap.
The resulting phase difference includes both travel through the film and any phase reversals at reflection.
For example, the colors of a soap film change with thickness because different wavelengths are reinforced.
Thiol
Chemistry · Definition
A thiol contains an S–H group attached to a carbon-containing structure.
Its bonding and properties differ from those of the corresponding alcohol, so replacing oxygen with sulfur does not preserve every interaction.
For example, ethanethiol is the sulfur-containing analogue of ethanol.
Threshold frequency
Physics 2 · Definition
A threshold frequency is the lowest light frequency that meets a surface’s work-function requirement in the single-photon photoelectric model.
Below it, increasing intensity alone does not produce emitted electrons in that model.
For example, a material with a larger work function requires higher-frequency photons to reach the emission threshold.
Titrant
Chemistry · Definition
A titrant is the reagent added in measured amounts during a titration.
Its concentration and reaction stoichiometry connect the delivered volume to the analyte amount.
For example, a standardized base solution can be the titrant for an unknown acid.
Titration
Chemistry · Definition
Titration measures an analyte by its reaction with a measured amount of another reagent.
The balanced reaction and the observed endpoint must be connected to the equivalence condition.
For example, an acid's amount can be inferred from a measured volume of a base with known concentration.
Titration endpoint
Chemistry · Definition
A titration endpoint is the observed signal used to stop or interpret a titration.
It approximates the equivalence point but can differ because of the chosen indicator or measurement conditions.
For example, an indicator's persistent color change may mark an endpoint near the steep region of a titration curve.
Torque
Physics 1 · Physics 2 · Definition
Torque describes a force’s turning effect about a chosen axis or point.
Its magnitude is force times the perpendicular lever arm, and its direction or sign describes the rotation tendency.
For example, pushing near the handle of a door produces more torque than the same perpendicular push near the hinge.
Total distance traveled
Calculus AB · Calculus BC · Definition
Total distance traveled adds the lengths of every part of a motion path without canceling opposite directions.
In one dimension it is the integral of speed, |v(t)|, over the time interval.
For example, moving 5 meters right and then 2 meters left gives total distance 7 meters.
When velocity changes sign, split the integral at those sign changes or integrate its absolute value correctly.
Total internal reflection
Physics 2 · Definition
Total internal reflection occurs when light travels toward a lower-index medium at an incidence angle above the critical angle.
Both the higher-to-lower index direction and the angle condition are required.
For example, light can remain guided inside an optical fiber by repeated total internal reflection at its boundary.
Transition state
Chemistry · Definition
A transition state is the high-energy configuration at a barrier along a reaction pathway.
It is not a stable intermediate that accumulates as a separately observable species in the basic model.
For example, the top of an elementary step's energy-profile peak represents its transition state.
Transverse wave
Physics 2 · Definition
In a transverse wave, the disturbance is perpendicular to the direction of propagation.
The disturbance direction and the energy-transfer direction must therefore be distinguished.
For example, a rope pulse can travel horizontally while rope segments move up and down.
Trapezoidal rule
Calculus AB · Calculus BC · Definition
The trapezoidal rule approximates a definite integral using a straight segment between each pair of adjacent endpoint values.
Each piece contributes its width times the average of its two endpoint heights.
For example, one trapezoid for x² on [0,2] has width 2 and heights 0 and 4, giving 4.
For a concave-up function the trapezoidal sum overestimates the integral, while for a concave-down function it underestimates it.
Treatment
Statistics · Definition
A treatment is a specific condition imposed on an experimental unit.
In a multifactor experiment, a treatment is a combination of factor levels.
For example, low fertilizer with eight hours of light is one treatment combination.
Tree diagram
Statistics · Definition
A probability tree represents successive stages of a chance process using branches.
Multiplying the relevant conditional probabilities along a path gives that path’s probability.
For example, a two-draw tree can show how a bag’s second-draw probabilities change when the first item is not replaced.
Trigonometric identity
Calculus AB · Calculus BC · Definition
A trigonometric identity is an equality that holds for every input where both sides are defined.
Identities let you rewrite expressions without changing their values on that shared domain.
For example, sin²θ+cos²θ=1 follows from the unit circle.
Any division used while rewriting an identity can introduce domain restrictions.
Trigonometric substitution
Calculus AB · Calculus BC · Definition
Trigonometric substitution replaces the variable of an integral with a trigonometric expression to simplify a radical.
For example, x=a sin θ can turn a²−x² into a² cos²θ when a>0.
The differential, bounds, and chosen angle interval must change consistently.
The angle interval also determines signs when taking square roots, so √(cos²θ) cannot always be replaced by cos θ.
Two-sample t procedure
Statistics · Definition
A two-sample t procedure compares population means using two independent samples or groups.
The usual unpooled procedure estimates each group’s variance separately, and paired data require a different analysis of pairwise differences.
For example, independently sampled travel times from two schools can estimate the difference between their population means.
Two-sample z procedure for proportions
Statistics · Definition
A two-sample z procedure uses a Normal approximation to compare two population proportions from independent groups.
The equality test pools proportions for its null standard error, while the usual confidence interval uses separate sample proportions.
For example, it can compare preference proportions from independently sampled students in two schools when its conditions hold.
Two-sided test
Statistics · Definition
A two-sided test seeks evidence for departures in either direction from the null claim.
For symmetric z or t tests, extreme evidence appears in both tails of the reference distribution.
For example, a not-equal alternative asks whether a population mean is either lower or higher than a stated value.
Two-way table
Statistics · Definition
A two-way table organizes counts for combinations of two categorical variables.
The cells show combinations, while row and column totals summarize each variable separately.
For example, a table can classify students by grade level and whether they walk to school.
Type I error
Statistics · Definition
A Type I error occurs when a test rejects a true null hypothesis.
The test’s significance level controls its probability under the null model.
For example, a study might report a treatment difference even though the population treatment effects are equal.
Type II error
Statistics · Definition
A Type II error occurs when a test fails to reject a false null hypothesis.
Its probability depends on a specified true alternative and the test’s design.
For example, a study might fail to detect a real treatment difference when the sample is too small.
Unbiased estimator
Statistics · Definition
An unbiased estimator has a sampling-distribution mean equal to the parameter being estimated.
Individual estimates can still differ substantially from the parameter, so unbiased does not mean perfectly accurate.
For example, a properly sampled mean can be unbiased even when small samples produce highly variable estimates.
Undercoverage
Statistics · Definition
Undercoverage occurs when some groups in the target population have little or no opportunity to enter the sample.
A large sample does not remove bias caused by an incomplete sampling frame.
For example, using only a landline directory misses people who have no listed landline.
Unimodal distribution
Statistics · Definition
A unimodal distribution has one main peak.
The peak marks a region where observations or probability are especially concentrated.
For example, a histogram of heights might have one prominent peak around the most common height interval.
Unit circle
Calculus AB · Calculus BC · Definition
The unit circle is the circle of radius 1 centered at the origin.
At angle θ measured from the positive horizontal axis, its point has coordinates (cos θ, sin θ).
For example, θ=π/2 gives the point (0,1), so cosine is 0 and sine is 1.
The circle also helps track signs and periodic behavior of trigonometric functions.
Unit vector
Physics 1 · Physics 2 · Definition
A unit vector has magnitude one and specifies a direction.
Multiplying it by a signed component gives the vector contribution along that direction.
For example, the Cartesian unit vectors identify the positive x, y, and z directions.
Valence electron
Chemistry · Definition
Valence electrons are the electrons most directly involved in an atom's bonding and chemical behavior.
For main-group atoms, they occupy the highest occupied principal shell.
For example, neutral chlorine has seven valence electrons, helping explain its tendency to form a chloride ion.
Vaporization
Chemistry · Definition
Vaporization is a phase change from liquid to gas.
It requires energy to separate particles against their attractive interactions without ordinarily breaking the molecules' internal bonds.
For example, boiling liquid water produces water vapor while preserving the water molecules.
Variance
Statistics · Definition
Variance is the square of the standard deviation.
It measures spread in squared units, with the precise averaging rule depending on whether it describes a sample or a probability distribution.
For example, a standard deviation of 3 centimeters corresponds to a variance of 9 square centimeters.
Vector
Physics 1 · Physics 2 · Chemistry · Definition
A vector is a quantity with a magnitude and a direction, represented by an arrow or by components along chosen axes.
Vectors add by adding corresponding components rather than simply adding their magnitudes.
For example, an eastward displacement followed by a northward displacement produces a resultant pointing northeast.
The zero vector has zero magnitude and no unique direction.
Vector component
Physics 1 · Physics 2 · Definition
A vector component describes the part of a vector along a chosen axis.
Its signed value changes when the axes change, even though the physical vector remains the same.
For example, a diagonal launch velocity can be resolved into horizontal and vertical components that share the same elapsed time.
Vector in calculus
Calculus BC · Definition
A vector records a magnitude and direction and can be represented by ordered components.
In planar motion, the components describe changes or rates along the horizontal and vertical axes.
For example, velocity ⟨3,4⟩ has speed 5 because its magnitude is √(3²+4²).
A vector-valued function lets those components vary with a parameter.
Vector-valued function
Calculus BC · Definition
A vector-valued function assigns a vector to each allowed input.
Its component functions can represent a moving point’s coordinates.
For example, r(t)=⟨cos t,sin t⟩ traces a circle, and its derivative is ⟨−sin t,cos t⟩.
Derivatives and integrals are computed component by component, while speed is the magnitude of the velocity vector.
Velocity
Physics 1 · Physics 2 · Definition
Velocity describes how position changes with time and includes direction.
Instantaneous velocity describes one moment, while average velocity is displacement divided by elapsed time.
For example, a car turning a corner can change velocity while maintaining the same speed.
Velocity in calculus
Calculus AB · Calculus BC · Definition
In one-dimensional motion, velocity is the derivative of position with respect to time.
Its sign records direction relative to the chosen positive axis.
For example, s(t)=t²−4t meters gives velocity 2t−4 meters per second.
Speed is the magnitude of velocity and therefore cannot be negative.
Venn diagram
Statistics · Definition
A Venn diagram represents sets or events with regions inside a sample-space boundary.
Overlapping regions show outcomes shared by two events.
For example, overlapping circles for red cards and kings identify the red kings.
Vertical asymptote
Calculus AB · Calculus BC · Definition
A vertical asymptote x=c occurs when function values tend to positive or negative infinity as x approaches c from at least one side.
Being unbounded near c is insufficient if the values oscillate without tending to either signed infinity.
For example, 1/x has vertical asymptote x=0, tending to positive infinity from the right and negative infinity from the left.
A zero denominator alone does not prove an asymptote because cancellation can instead leave a removable hole.
Vertical shift
Calculus AB · Calculus BC · Definition
A vertical shift adds the same constant to every output of a function.
The graph of f(x)+k is shifted upward by k when k is positive and downward when k is negative.
For example, sin x+3 has midline y=3.
The shift changes the midline without changing a sinusoid’s amplitude or period.
Vertical tangent
Calculus AB · Calculus BC · Definition
A vertical tangent is a tangent line parallel to the vertical axis, with no finite slope.
A graph can have a vertical tangent even though its derivative does not exist as a finite real number at that input.
For example, y=∛x has the vertical tangent x=0 at the origin as its nearby slopes tend to positive infinity.
This case is separate from the finite-slope tangent approximation available at a differentiable point.
Virtual image
Physics 2 · Definition
A virtual image is located where backward extensions of outgoing rays appear to meet.
The rays do not physically converge at that location, so placing a screen there does not capture that image.
For example, a plane mirror produces an image that appears behind the mirror.
Viscosity
Physics 1 · Physics 2 · Definition
Viscosity measures a fluid’s resistance to relative motion between neighboring layers.
Viscous effects can dissipate mechanical energy and alter the velocity profile of flowing fluid.
For example, honey flows more reluctantly than water under comparable conditions because it is more viscous.
Voltmeter
Physics 2 · Definition
A voltmeter measures potential difference between two nodes.
It is connected in parallel and ideally has very high resistance so it draws negligible current.
For example, placing its probes on opposite ends of a resistor measures that resistor’s voltage drop.
Voluntary response sample
Statistics · Definition
A voluntary response sample consists of people who choose to respond to an open invitation.
People with strong opinions may participate more often than others, creating selection bias.
For example, an unrestricted online poll can overrepresent those especially interested in the issue.
VSEPR
Chemistry · Definition
VSEPR is a model for predicting molecular geometry from electron domains around a central atom.
Bonding domains and lone pairs arrange to reduce repulsion, and a multiple bond counts as one domain for the basic geometry classification.
For example, water has four electron domains around oxygen but a bent molecular shape.
Washer method
Calculus AB · Calculus BC · Definition
The washer method uses circular slices with a central hole to compute volume.
A slice with outer radius R and inner radius r has area π(R²−r²).
For example, rotating the region 1≤y≤2 on a horizontal interval around the horizontal axis gives washer area 3π.
Both radii are distances from the rotation axis and must be expressed using the same slice variable.
Wave
Physics 2 · Definition
A wave is a propagating disturbance that can transfer energy without transporting the medium as a whole.
Mechanical waves require a material medium, while electromagnetic waves can propagate through vacuum.
For example, a pulse travels along a rope while individual rope segments move locally around their equilibrium positions.
Wave intensity
Physics 2 · Definition
Wave intensity is the time-averaged energy-transfer rate per area perpendicular to propagation.
It is measured in watts per square meter and differs from total power because area matters.
For example, spreading the same sound power over a larger area reduces the intensity there.
Wave interference
Physics 2 · Definition
Wave interference is the combination of overlapping wave disturbances.
In a linear medium, the signed disturbances add at each location and time without permanently destroying the passing waves.
For example, two overlapping upward rope pulses temporarily make a larger upward disturbance.
Wave node
Physics 2 · Definition
A wave node is a location where the specified oscillating quantity has zero oscillation amplitude in a standing-wave pattern.
Displacement nodes and pressure nodes in a sound standing wave need not occur at the same positions, so the quantity must be named.
For example, the fixed ends of an ideal vibrating string are displacement nodes.
Wave polarization
Physics 2 · Definition
Wave polarization specifies the orientation of a transverse wave’s oscillation.
For light, an ideal linear polarizer selects an electric-field component along its transmission direction.
For example, crossed ideal linear polarizers block light transmitted through the first polarizer.
Wave–particle duality
Physics 2 · Definition
Wave–particle duality describes quantum observations that require both wave-like and particle-like features in appropriate models.
Localized detection events can build an interference pattern, so neither an ordinary classical particle picture nor a classical wave picture is sufficient alone.
For example, electrons detected one at a time can collectively form a double-slit interference pattern.
Wavefront
Physics 2 · Definition
A wavefront is a surface joining points of equal wave phase at a given time.
Rays are perpendicular to wavefronts in the uniform isotropic ray model used here.
For example, a distant source can be approximated by nearly plane wavefronts over a small region.
Wavelength
Physics 2 · Definition
Wavelength is the spatial distance between neighboring points at the same phase of a periodic wave.
It connects propagation speed and frequency through speed equal to frequency times wavelength.
For example, neighboring crests of a sinusoidal water wave are separated by one wavelength.
Wavenumber
Physics 2 · Definition
Angular wavenumber measures the change of wave phase per unit distance.
In the sinusoidal convention used in the source reading, it equals two pi divided by wavelength and is measured in radians per meter.
For example, shortening a wave’s wavelength increases its angular wavenumber.
Weak acid or base
Chemistry · Definition
A weak acid or base reacts only partially with water and establishes an equilibrium.
Its equilibrium constant describes strength, while analytical concentration describes how much was prepared per volume.
For example, acetic acid leaves both acid molecules and its conjugate base in water.
Weight
Physics 1 · Physics 2 · Definition
Weight is the gravitational force exerted on an object by a specified attracting body.
Near Earth’s surface its magnitude is approximately mass times the local gravitational field strength, and it is measured in newtons.
For example, an astronaut has less weight on the Moon while retaining the same mass.
Wien’s displacement law
Physics 2 · Definition
Wien’s displacement law relates a blackbody spectrum’s peak wavelength to its kelvin temperature.
The peak wavelength is inversely proportional to temperature when the spectrum is expressed per unit wavelength.
For example, a hotter blackbody peaks at a shorter wavelength, though it still emits across a broad range.
Work function
Physics 2 · Definition
A work function is the minimum energy needed to remove an electron from the specified material surface.
For single-photon photoemission, the photon energy first pays this energy requirement before contributing to the emitted electron’s kinetic energy.
For example, two different surfaces can have different threshold frequencies because their work functions differ.
Z-score
Statistics · Definition
A z-score expresses a value’s distance from its distribution’s mean in standard-deviation units.
It is found by subtracting the mean from the value and dividing by the standard deviation.
For example, a z-score of −1.5 places the value 1.5 standard deviations below the mean.
Z-table
Statistics · Definition
A z-table lists probabilities or areas for the standard Normal distribution.
Tables use different conventions, so the headings must be checked to identify which area is listed.
For example, a cumulative-left table gives the area to the left of each z-value.