Key definitions

A

Acceleration

The rate of change of velocity with time. — Kinematics with variable acceleration

Actual significance level

The probability, calculated assuming the null hypothesis is true, of landing in the critical region: the test's real false-alarm rate. — Hypothesis testing with the binomial

Amplitude

Half the difference between the maximum and minimum values of a periodic function. — Trigonometric graphs and equations

Apollonius locus

The set of points whose distances from two fixed points are in a fixed ratio other than 1. It is a circle, not a line; equal distances give the perpendicular bisector instead. — Further loci and regions in the Argand diagram

Arithmetic sequence

A sequence in which the difference between consecutive terms is constant, the common difference d. — Arithmetic series

Asymptote

A line the curve approaches arbitrarily closely without meeting, as x or y grows without bound. — Graphs, proportion and transformations

B

Bellman's principle of optimality

Any part of an optimal path is itself optimal between its own two ends. That is what licenses working backwards from the final stage, quoting values already found. — Dynamic programming

Bezout's identity

The highest common factor of a and b can be written as ax + by for integers x and y, found by unwinding the Euclidean algorithm. When that factor is 1, x is the inverse of a modulo b. — The Euclidean algorithm and Bezout's identity

Bin packing lower bound

The total of the items divided by the bin size, rounded up. If an algorithm achieves it the packing is optimal; falling short proves nothing either way. — Algorithms, sorting and bin packing

Binomial coefficient

The number of ways of choosing r objects from n, written nCr, which is the coefficient of the corresponding term in the expansion of (a + b)ⁿ. — The binomial expansion

C

Capacity of a cut

The total capacity of the arcs directed from the source side to the sink side only. Arcs crossing back towards the source contribute nothing at all. — Flows in networks: cuts and capacity

Cartesian form

The equation of a curve in x and y alone, obtained from parametric equations by eliminating the parameter. — Parametric equations

Cayley-Hamilton theorem

Every square matrix satisfies its own characteristic equation, which turns any power of it into a linear combination of lower powers and gives the inverse without cofactors. — Diagonalisation and the Cayley-Hamilton theorem

Change of sign

If a continuous function takes opposite signs at the ends of an interval, it has a root inside; continuity is what makes the argument work. — Locating roots and iteration

Choosing the acceleration

Use v dv/dx when the force depends on position, dv/dt when it depends on time. The constant-acceleration equations are not available at all, whatever average is taken. — Newton's laws with a variable force

Coefficient of friction

The constant μ for which the maximum friction available between two surfaces is μ times the normal reaction. — Friction and inclined planes

Collinear points

Points lying on one straight line, shown by two of the joining vectors being parallel and sharing a point. — Vectors in three dimensions

Common difference

The constant amount added to each term of an arithmetic sequence to reach the next. — Arithmetic series

Common ratio

The constant factor multiplying each term of a geometric sequence to reach the next. — Geometric series

Completed square form

A quadratic written as a(x + p)² + q, which exhibits the vertex at (−p, q) and so the maximum or minimum value directly. — Quadratic functions

Completing a vertical circle

On a string the tension must stay non-negative, so v² ≥ gr at the top and u² ≥ 5gr at the bottom. On a rod, which can push, the particle need only arrive at the top with v > 0. — Motion in a vertical circle

Composite function

The function fg, meaning apply g first and then f, defined wherever the output of g lies in the domain of f. — Functions, inverses and the modulus

Conditional probability

The probability of A given B, that is the probability of A within the cases where B has occurred. — Conditional probability

Conjecture

A statement believed to be true but not yet proved. — The structure of proof: deduction and exhaustion

Conservation of momentum

In a collision the internal forces are equal and opposite for the same time, so the impulses cancel and the total momentum before equals the total after. It holds whatever happens to the energy. — Momentum and impulse

Convergent geometric series

A geometric series has a sum to infinity exactly when |r| < 1, and that sum is a/(1 − r). — Geometric series

Counterexample

A single specific case for which a general statement fails, which is enough to disprove the statement. — Disproof and proof by contradiction

Critical activity

One whose earliest and latest start times agree, so it has zero float. The critical activities form a continuous path whose durations total the project duration. — Critical path analysis

Critical region

The set of outcomes for which the null hypothesis is rejected, chosen so that its probability under the null hypothesis is at most the significance level. — Hypothesis testing with the binomial

Critical values

The values at which a quadratic inequality changes sign, that is the roots of the corresponding equation; the solution is read off between or outside them according to the direction of the inequality. — Simultaneous equations and inequalities

D

Decreasing function on an interval

A function whose derivative is negative throughout the interval. — Tangents, turning points and curve behaviour

Definite integral

The signed area between a curve and the x-axis over an interval, evaluated as the change in any antiderivative across that interval. — Definite integrals and areas

Derivative

The instantaneous rate of change of a function with respect to its variable, equal to the gradient of the tangent at that point. — Differentiating powers of x

Differential equation

An equation relating a function to one or more of its derivatives. — Rates of change and building differential equations

Differentiation from first principles

The derivative of f(x) is the limit as h tends to zero of (f(x + h) − f(x))/h, provided the limit exists. — The derivative from first principles

Dijkstra's rule

Make permanent the smallest temporary label anywhere in the network, not the cheapest edge from where you are. Read the route back by differencing final labels against edge weights. — Shortest paths: Dijkstra and Floyd

Discriminant

For ax² + bx + c, the quantity b² − 4ac, whose sign gives the number of real roots: positive two, zero one repeated, negative none. — Quadratic functions

Displacement

The change in position, a vector, as distinct from distance travelled, which is a scalar and never decreases. — Kinematics with constant acceleration

Domain

The set of inputs on which a function is defined. — Functions, inverses and the modulus

Dominance

A row is dominated when another is at least as good in every column; a column is dominated when another is at least as small in every row. Dominated options are deleted outright, often reducing a game to a two by two. — Game theory: play safe and stable solutions

E

Equilibrium

A body is in equilibrium when the resultant force on it is zero, so it stays at rest or moves with constant velocity. — Statics of a particle

Eulerian and semi-Eulerian

No odd nodes means a closed trail exists using every edge once; exactly two odd nodes means such a trail exists but must start and end at them; more than two means neither. — Graphs: order, Eulerian paths and planarity

Expected monetary value

A chance node takes the probability-weighted average of its branches, a decision node takes the best available, working back to the first decision. Costs come off along the branch where they are met. — Decision analysis

Exponential growth and decay

A quantity changing at a rate proportional to its current size, giving growth when the constant is positive and decay when it is negative. — Exponential functions and e

F

Flow-augmenting path

A route from source to sink using spare capacity forwards or existing flow backwards. It carries the smallest of those numbers along it, and at least one arc becomes saturated. — Maximum flow and the labelling procedure

Frequency density

Frequency divided by class width, plotted as the height of a histogram bar so that area rather than height represents frequency. — Representing and interpreting data

Function

A rule that assigns to each element of its domain exactly one element of its range. — Functions, inverses and the modulus

G

Geometric sequence

A sequence in which the ratio between consecutive terms is constant, the common ratio r. — Geometric series

Gradient of a line

The increase in y per unit increase in x, constant along a straight line. — Straight lines

Group

A set with a binary operation satisfying closure, associativity, the existence of an identity, and an inverse for every element. Commutativity is not required. — Groups and their axioms

H

Harmonic form

a sin θ + b cos θ written as R sin(θ + α), which exposes the maximum R and minimum −R without differentiating. — Compound angles and the harmonic form

Hungarian reduction

Subtract each row's minimum from that row, then each column's from that column. Every complete allocation loses the same total, so the cheapest one is unchanged and the zeros mark the bargains. — Allocation and the Hungarian algorithm

I

Identity

An equation true for every value of the variable for which both sides are defined, as distinct from an equation true only at particular values. — Trigonometric modelling

If and only if

Two statements are connected by if and only if when each implies the other, so a proof of an equivalence has two directions to establish. — The structure of proof: deduction and exhaustion

Implicit differentiation

Differentiating an equation in x and y as it stands, applying the chain rule to every term in y, rather than first solving for y. — Implicit and parametric differentiation

Improper integral

An integral with an infinite limit or an integrand unbounded at a limit, defined as the limit of an ordinary integral, and said to converge when that limit is finite. — Mean values and improper integrals

Impulse in two dimensions

The i components and the j components each obey I = mv − mu on their own. Take magnitudes only at the end: a change of direction at constant speed still needs an impulse. — Impulse and momentum as vectors

Increasing function on an interval

A function whose derivative is greater than or equal to zero throughout the interval, so its value never falls as x increases. — Tangents, turning points and curve behaviour

Indefinite integral

The family of all functions with a given derivative, written with an arbitrary constant because the constant is lost on differentiating. — Integration as antidifferentiation

Independent events

Events A and B are independent when P(A ∩ B) = P(A) × P(B); equivalently, when P(A|B) = P(A), so B carries no news about A. — Probability and Venn diagrams

Integration by parts

The reverse of the product rule: the integral of u dv/dx equals uv minus the integral of v du/dx. — Integration by substitution and by parts

Interpolation and extrapolation

Using a regression line inside the range of the data is interpolation and is reliable; using it outside that range is extrapolation and is not. — Correlation and regression

Interquartile range

The upper quartile minus the lower quartile, a measure of spread unaffected by extreme values. — Measures of location and spread

Invariant line

A line whose every point maps to a point of the same line under the transformation. Points may move along it; none leaves it. — Systems of equations and invariance

K

Kinetic energy in an impact

Momentum is always conserved; kinetic energy is lost unless e = 1. Find the loss by adding ½mv² for both spheres before and after and subtracting. — Direct impact and Newton's law of restitution

L

L'Hospital's rule

For an indeterminate form 0/0 or ∞/∞, the limit of f/g equals the limit of f'/g', provided that second limit exists. — Limits and L'Hospital's rule

Lagrange's theorem

The order of any subgroup of a finite group divides the order of the group, so the order of every element divides it too. — Subgroups, Lagrange's theorem and isomorphism

Lamina against framework

A uniform lamina weights each piece by its area at its own centroid, with a negative area for anything removed. A framework weights each rod by its length at its own midpoint, so the two differ for the same shape. — Centres of mass of plane figures and frameworks

Limiting equilibrium

The state in which an object is still in equilibrium but on the point of slipping, so friction has reached its maximum value μR. — Friction and inclined planes

Linearising a model

Taking logarithms of a power or exponential model so that the relationship plots as a straight line, whose gradient and intercept give the constants. — Log graphs and exponential models

Logarithm

log to base a of x is the power to which a must be raised to give x: if a^y = x then y equals log base a of x, for a > 0, a ≠ 1 and x > 0. — Logarithms and their laws

M

m + n − 1

The number of cells a solution to an m by n transportation problem must use, which is exactly enough to fix the shadow costs. Fewer means degeneracy, fixed by recording a zero allocation. — Transportation problems

Maximising an allocation

Subtract every entry from the largest entry in the matrix, then minimise as usual. The biggest profit becomes the smallest cost, so the allocation found is the one of greatest total. — Allocation and the Hungarian algorithm

Minimum spanning tree

Connects all n nodes with n − 1 edges and no cycle, at least total weight. Prim grows from a node; Kruskal works down the sorted edges rejecting cycles. Both give the same total. — Minimum spanning trees: Prim and Kruskal

Moment of a force

The turning effect of a force about a point: the magnitude of the force times the perpendicular distance from the point to its line of action. — Moments

Mutually exclusive

Two events are mutually exclusive when they cannot both occur, so the probability of their intersection is zero. — Probability and Venn diagrams

N

Newton's first law

A body remains at rest or moves with constant velocity unless acted on by a resultant force. — Forces and Newton's laws

Newton's second law

The resultant force on a body equals its mass times its acceleration, and acts in the direction of that acceleration. — Forces and Newton's laws

Newton's third law

If one body exerts a force on another, the second exerts a force on the first that is equal in magnitude and opposite in direction. — Forces and Newton's laws

Normal reaction

The component of the contact force perpendicular to the surface. — Connected particles and pulleys

North-west corner method

Fill the top-left cell as far as supply and demand allow, cross off whichever runs out, repeat. It never looks at a cost, so the plan is usually poor: its job is to be feasible, not good. — Transportation problems

Null and alternative hypotheses

The null hypothesis is the statement about the population being tested; the alternative is what is accepted if the evidence is strong enough against it. — Hypothesis testing with the binomial

O

Oblique impact

A smooth surface leaves the parallel component untouched and multiplies the perpendicular one by −e. The outgoing path therefore always hugs the surface more closely than the incoming one. — Oblique impact and impact with a smooth surface

One-tailed and two-tailed

A one-tailed test looks for a change in a stated direction; a two-tailed test looks for a change in either, and splits the significance level between the two tails. — Hypothesis testing: correlation and the normal

One-to-one

A function is one-to-one when distinct inputs give distinct outputs, which is exactly the condition for an inverse function to exist. — Functions, inverses and the modulus

Order of an algorithm

How the number of operations grows with the size n of the problem. An order n² algorithm takes four times as long on a list twice as long. — Algorithms, sorting and bin packing

Outlier

A value far enough from the rest to be treated separately, judged against a stated rule such as more than 1.5 interquartile ranges beyond a quartile. — Representing and interpreting data

P

Parameter

A third variable in terms of which x and y are each given, so that letting it vary traces the curve. — Parametric equations

Partial fractions

A single algebraic fraction rewritten as a sum of fractions with simpler denominators, one for each factor of the original denominator. — Partial fractions

Particle

A body modelled as having mass but no size, so that its rotation and any variation across it can be ignored. — Modelling, quantities and units

Particular solution

The single solution of a differential equation picked out from the general family by a given condition. — Solving differential equations

Period

The smallest positive interval after which a periodic function repeats. — Trigonometric graphs and equations

Periodic sequence

A sequence that repeats after a fixed number of terms, that number being its period. — Sequences and sigma notation

Perpendicular gradients

Two non-vertical lines are perpendicular exactly when the product of their gradients is −1. — Straight lines

Piecewise function

A function given by different expressions on different parts of its domain. — Functions in modelling

Play safe and stable

The row player takes the row whose worst entry is best, the maximin; the column player takes the column whose largest is smallest, the minimax. The game is stable exactly when the two agree, and that number is its value. — Game theory: play safe and stable solutions

Point of inflection

A point where the curve changes its sense of concavity, so the second derivative changes sign. — Tangents, turning points and curve behaviour

Population and sample

The population is every member of the group under study; a sample is the subset actually measured. — Sampling and the large data set

Position vector

The vector from a fixed origin to a point, so its components are the coordinates of that point. — Vectors in two dimensions

Positive skew

Mode below median below mean, with the long tail running to the right. Reversing the order gives negative skew; all three equal means symmetric. — Mean, variance and skewness of continuous variables

Power of a test

The probability of correctly rejecting the null hypothesis at a stated alternative, equal to one minus the probability of a Type II error there. — The quality of tests

Principal argument

The angle a complex number's position vector makes with the positive real axis, measured anticlockwise, in the interval (−π, π]. — Modulus, argument and loci

Principal value

The single output an inverse trigonometric function returns, chosen from a restricted range so that the inverse is a function at all. — Reciprocal and inverse trigonometric functions

Projectile

A body moving freely under gravity alone, so its horizontal velocity is constant and its vertical acceleration is g downwards. — Projectiles

Proof by contradiction

A proof that assumes the negation of the statement and derives a contradiction, so the negation is false and the statement is true. — Disproof and proof by contradiction

Proof by exhaustion

A proof that checks every case in a complete list of cases, so it is valid only when the list is genuinely exhaustive. — The structure of proof: deduction and exhaustion

R

Radian

The angle subtended at the centre of a circle by an arc equal in length to the radius. — Radians, arcs and small angles

Range

The set of outputs a function actually takes over its domain. — Functions, inverses and the modulus

Rationalising the denominator

Rewriting a quotient so that no surd remains in the denominator, by multiplying above and below by a suitable factor. — Indices and surds

Recurrence relation

A rule defining each term of a sequence from earlier terms, together with a starting value. — Sequences and sigma notation

Residual

The observed y minus the value the regression line predicts at the same x: a vertical gap. Least squares minimises the sum of their squares, and they always add to zero. — Least squares regression and residuals

S

Sampling frame

The list of the population from which a sample is drawn; a member missing from the frame can never be sampled. — Sampling and the large data set

Scalar and vector

A scalar has magnitude only; a vector has magnitude and direction. — Vectors in two dimensions

Separating the variables

Rearranging a differential equation so each variable appears on its own side with its own differential, after which both sides are integrated. — Solving differential equations

Significance level

The probability of rejecting the null hypothesis when it is in fact true, fixed before the data are seen. — Hypothesis testing with the binomial

Simple random sample

A sample in which every member of the population is equally likely to be chosen, and every possible sample of that size equally likely. — Sampling and the large data set

Slack, surplus and artificial

A ≤ constraint gains a slack variable; a ≥ constraint loses a surplus variable and then gains an artificial one so that a starting solution exists at all. — Linear programming: formulation and graphical solution

Small angle approximations

For θ in radians and small, sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − θ²/2; the results are false if θ is in degrees. — Radians, arcs and small angles

Smooth and light

Smooth means friction is modelled as zero; light means mass is modelled as zero, so tension is the same throughout a light string. — Modelling, quantities and units

Solving a mixed game

Play the first row with probability p, write the expected pay-off against each column as a line in p, and take the highest point of the lower boundary. Equate the two lines meeting there for p and the value. — Mixed strategies

Staircase and cobweb diagrams

Sketches of an iteration on the curve and the line y = x, converging in steps to a root or spiralling about it. — Locating roots and iteration

Standard deviation

The root mean square deviation from the mean, in the same units as the data. — Measures of location and spread

Stationary point

A point where the derivative is zero. — Tangents, turning points and curve behaviour

Surd

An irrational root left in exact form rather than written as a decimal. — Indices and surds

T

Tangent to a circle

A line meeting the circle exactly once; it is perpendicular to the radius at the point of contact. — Circles

The ambiguous case

When the sine rule is used to find an angle, two angles in the range have the same sine, so both the acute and the obtuse possibility must be tested against the triangle. — Triangles and the sine and cosine rules

The binomial conditions

A fixed number of independent trials, each with two outcomes and the same probability of success throughout. — The binomial distribution

The chain rule

For a function of a function, dy/dx = (dy/du)(du/dx). — The product, quotient and chain rules

The correlation test

Hypotheses are about the population value ρ or ρs, never the sample r. Critical values fall as n grows, and the product moment version assumes a bivariate normal population. — Testing a correlation coefficient

The equation of a circle

(x − a)² + (y − b)² = r² is the circle of centre (a, b) and radius r. — Circles

The factor theorem

(x − a) is a factor of the polynomial f(x) if and only if f(a) = 0. — Polynomials and the factor theorem

The limit of a sum

The definite integral as the limit of the total area of rectangles under the curve as their width tends to zero. — Areas, parametric curves and the limit of a sum

The minimum lines test

If the fewest horizontal and vertical lines covering every zero equals the matrix size, an allocation of zeros exists. If fewer, subtract the smallest uncovered entry from all uncovered ones and add it to every doubly covered one. — Allocation and the Hungarian algorithm

The Newton-Raphson method

An iteration taking the next estimate as the point where the tangent at the current one crosses the x-axis, which fails if that derivative is zero. — Newton-Raphson and the trapezium rule

The normal distribution

A continuous distribution, symmetric about its mean, in which about 95 per cent of values lie within two standard deviations of the mean. — The normal distribution

The number e

The base for which the gradient of y = e^x equals its own value at every point. — Exponential functions and e

The product moment correlation coefficient

A measure of linear association between −1 and 1, where 0 indicates no linear relationship and never that the variables are unrelated. — Correlation and regression

The remainder theorem

When a polynomial f(x) is divided by (x − a), the remainder is f(a). — Polynomials and the factor theorem

The Simplex rules

Pivot on the most negative entry in the objective row, choosing the row by the smallest non-negative ratio of value to pivot-column entry. Stop when no negative entries remain. — The Simplex algorithm

The stepping-stone loop

From the entering cell, trace a closed route through cells in use, alternating plus and minus. The value moved is the smallest minus-cell allocation, that cell exits, and the cost falls by that value times the index. — The stepping-stone method

The trapezium rule

An estimate of a definite integral that replaces the curve by straight chords, overestimating where the curve is concave up and underestimating where it is concave down. — Newton-Raphson and the trapezium rule

The validity condition

The expansion of (1 + x)ⁿ for non-integer n is an infinite series that converges only for |x| < 1, so a valid answer states that range. — The general binomial expansion

Travelling salesman lower bound

Delete a node, find a minimum spanning tree on what is left, then add back the two shortest edges from the deleted node. Any tour at all gives an upper bound. — The travelling salesman problem

U

Uniform beam

A beam whose mass is evenly spread, so its whole weight can be modelled as acting at its midpoint. — Moments

Unit vector

A vector of magnitude 1, obtained from any non-zero vector by dividing it by its own magnitude. — Vectors in two dimensions

V

Velocity and speed

Velocity is the rate of change of displacement and is a vector; speed is its magnitude. — Kinematics with constant acceleration

W

Weight

The force of gravity on a body, of magnitude mg, directed vertically downwards. — Forces and Newton's laws

Work-energy principle

The total work done by all the forces equals the change in kinetic energy. The normal reaction never contributes, since it acts at right angles to the motion. — Work, energy and power