Maths (university) glossary
92 key terms from Maths (university), A to Z. Each is explained in one plain sentence and links to its free lesson.
Last updated
Prefer to learn in order? See every Maths (university) lesson.
A
- Alternating seriesSequences and series
- A series whose terms switch sign.
- AntiderivativeIntegration
- A function F whose derivative is f.
- Area between curvesIntegration
- Integrate the top curve minus the bottom curve over the interval where they enclose a region.
- Augmented matrixVectors, matrices and linear systems
- A matrix that stores a system's coefficients and its right-hand sides together, separated by a bar.
- Average valueIntegration
- The average value of f on [a, b] is the integral of f over [a, b], divided by the length b - a.
C
- Chain ruleDifferentiation
- To differentiate a function inside another function, differentiate the outer one (leaving the inside alone), then multiply by the derivative of the inside.
- Characteristic equationEigenvalues and eigenvectors
- The eigenvalues of A are the solutions of det(A - lambda I) = 0.
- Choosing u and dvIntegration techniques and uses
- A good choice makes the new integral simpler than the old one.
- Cofactor expansionDeterminants, inverses and vector spaces
- Compute a larger determinant by going along a row, multiplying each entry by the determinant of the smaller matrix left after crossing out its row and column.
- Complex eigenvaluesEigenvalues and eigenvectors
- Some real matrices have no real eigenvectors, because their characteristic equation has complex roots such as i and -i.
- ContinuityLimits and continuity
- A function is continuous at a if f(a) exists, the limit at a exists, and the two are equal.
- ContrapositiveLogic, sets and proof
- The implication 'if not Q then not P'.
- Critical pointApplications of derivatives
- A value of x where f'(x) = 0 or f'(x) does not exist.
- Cross productVectors, matrices and linear systems
- For two vectors in 3D, the cross product is a new vector at right angles to both.
D
- Definite integralIntegration
- The integral of f from a to b is the signed area between the curve and the x-axis.
- DerivativeDifferentiation
- The derivative of f at a is the limit of (f(a + h) - f(a)) / h as h tends to 0.
- DeterminantDeterminants, inverses and vector spaces
- A single number computed from a square matrix.
- DiagonalisationEigenvalues and eigenvectors
- If A has enough independent eigenvectors, then A = P D P^(-1).
- DifferentiabilityDifferentiation
- A function is differentiable at a if its derivative exists there.
- Direct proofLogic, sets and proof
- Start from the assumptions and move step by step, using definitions and known facts, until you reach the conclusion.
- Divergence testSequences and series
- If the terms of a series do not tend to 0, the series diverges.
- Dot productVectors, matrices and linear systems
- For vectors a and b, multiply matching entries and add.
E
- Eigenvector and eigenvalueEigenvalues and eigenvectors
- An eigenvector of A is a non-zero vector v with A v = lambda v.
- Epsilon-delta definitionLimits and continuity
- The limit of f(x) at a is L if, for every tolerance epsilon > 0, there is a distance delta > 0 so that x within delta of a gives f(x) within epsilon of L.
F
- Finding eigenvectorsEigenvalues and eigenvectors
- For each eigenvalue lambda, solve (A - lambda I) v = 0.
- First derivative testApplications of derivatives
- Check the sign of f' on each side of a critical point.
- Fundamental theorem of calculusIntegration
- If F' = f, then the integral of f from a to b equals F(b) - F(a).
G
- Gaussian eliminationVectors, matrices and linear systems
- Use row operations to create zeros below each leading entry, until the system is in staircase (echelon) form.
- Geometric seriesSequences and series
- A series where each term is a fixed multiple r of the one before.
I
- ImplicationLogic, sets and proof
- 'If P then Q'.
- Implicit differentiationDifferentiation
- When y is tangled up with x in an equation, differentiate both sides with respect to x and treat y as a function of x, using the chain rule.
- Improper integralIntegration techniques and uses
- An integral with an infinite limit, or a function that blows up inside the interval.
- InductionLogic, sets and proof
- To prove a claim for every whole number n, show it for n = 1 (base case).
- Injective and surjectiveLogic, sets and proof
- An injective function never sends two different inputs to the same output.
- Integration by partsIntegration techniques and uses
- The integral of u dv equals uv minus the integral of v du.
- Integration by substitutionIntegration
- Replace a tangled part with a new letter u so the integral becomes simple.
- Intermediate value theoremLimits and continuity
- If f is continuous on [a, b] and takes values with opposite signs at the ends, then it equals zero somewhere between them.
- Inverse matrixDeterminants, inverses and vector spaces
- The inverse of A, written A^(-1), satisfies A times A^(-1) = I.
L
- L'Hopital's ruleApplications of derivatives
- If a limit gives 0/0 or infinity/infinity, differentiate the top and the bottom separately and try the limit again.
- LimitLimits and continuity
- The limit of f(x) as x approaches a is the value f(x) gets closer and closer to.
- Limit at infinityLimits and continuity
- The value a function approaches as x grows without bound.
- Limit lawsLimits and continuity
- Limits of sums, differences, products and quotients behave as you would hope: take each limit, then combine.
- Linear approximationApplications of derivatives
- Near x = a, a curve looks like its tangent line.
- Linear independenceDeterminants, inverses and vector spaces
- Vectors are independent if none of them can be built from the others by scaling and adding.
- Linear systemVectors, matrices and linear systems
- A set of equations where each unknown appears only to the first power, such as x + y = 5 and x - y = 1.
- Linear transformationDeterminants, inverses and vector spaces
- A function on vectors that keeps addition and scaling intact.
M
- MatrixVectors, matrices and linear systems
- A rectangular table of numbers with rows and columns.
- Matrix multiplicationVectors, matrices and linear systems
- Each entry of the product is a row of the first matrix dotted with a column of the second.
- Matrix powersEigenvalues and eigenvectors
- With A = P D P^(-1), the power A^n equals P D^n P^(-1).
- Mean value theoremApplications of derivatives
- If f is continuous on [a, b] and differentiable inside, there is a point c between a and b where f'(c) equals the average slope.
N
- Newton's methodApplications of derivatives
- A way to find roots by repeating x_new = x - f(x)/f'(x).
- Number of solutionsVectors, matrices and linear systems
- A linear system has exactly one solution, no solution, or infinitely many.
- Numerical integrationIntegration techniques and uses
- When no antiderivative can be found, estimate the integral with the trapezoid rule or Simpson's rule, which use values of the function at several points.
O
- One-sided limitLimits and continuity
- A limit where x approaches a from only one side: from the left (x below a) or from the right (x above a).
- OptimisationApplications of derivatives
- Turn a real problem into a function of one variable, find its critical points, and pick the best one.
P
- p-seriesSequences and series
- The series of 1 / n^p converges if p > 1 and diverges if p <= 1.
- Partial fractionsIntegration techniques and uses
- Split a fraction with a factorised denominator into simpler fractions that are easy to integrate.
- Partial sumSequences and series
- The sum of the first n terms of a series.
- Power ruleDifferentiation
- The derivative of x^n is n x^(n-1).
- Power rule for integralsIntegration
- The integral of x^n dx is x^(n+1) / (n+1) + C, for n not equal to -1.
- Product ruleDifferentiation
- The derivative of u times v is u'v + uv'.
- Proof by contradictionLogic, sets and proof
- Assume the opposite of what you want to prove.
Q
- QuantifiersLogic, sets and proof
- 'For all' says a claim holds for every member of a set. 'There exists' says it holds for at least one.
- Quotient ruleDifferentiation
- The derivative of u over v is (u'v - uv') divided by v^2.
R
- Radius of convergenceSequences and series
- A power series (a sum of terms a_n x^n) converges for x with |x| < R, where R is the radius of convergence.
- Rank and dimensionDeterminants, inverses and vector spaces
- The rank of a matrix is the number of independent columns.
- Rate of changeDifferentiation
- The derivative with respect to time, or any other input, tells you how fast the quantity changes.
- Ratio testSequences and series
- Find the limit L of |a_(n+1) / a_n|.
- Related ratesApplications of derivatives
- When two or more quantities change with time and are linked by an equation, differentiate the equation with respect to time to link their rates.
- Riemann sumIntegration
- An estimate of area under a curve made by adding up thin rectangles.
S
- Second derivative testApplications of derivatives
- At a critical point, if f'' is positive the curve bends upward (a minimum).
- Separable differential equationIntegration techniques and uses
- An equation for dy/dx that can be rearranged to have all the y terms on one side and all the x terms on the other.
- Sequence limitSequences and series
- A sequence is a list of numbers a_1, a_2, a_3, and so on.
- Set operationsLogic, sets and proof
- A set is a collection of distinct objects.
- Signed area and net changeIntegration
- An integral adds area above the axis and subtracts area below it.
- Solving with the inverseDeterminants, inverses and vector spaces
- If A x = b and A is invertible, then x = A^(-1) b.
- Span and basisDeterminants, inverses and vector spaces
- The span of vectors is everything you can build from them by scaling and adding.
- Squeeze theoremLimits and continuity
- If a function is trapped between two functions that have the same limit, it must have that limit too.
- Standard derivativesDifferentiation
- Know these by heart: sin x gives cos x, cos x gives -sin x, e^x gives e^x, and ln x gives 1/x for x > 0.
- Standard integralsIntegration
- Reverse the standard derivatives: cos x integrates to sin x, sin x to -cos x, e^x to e^x and 1/x to ln|x|, each plus C.
- StatementLogic, sets and proof
- A sentence that is definitely true or definitely false, never both.
- Steady stateEigenvalues and eigenvectors
- For a Markov chain (columns of probabilities summing to 1), the long-run distribution is the eigenvector with eigenvalue 1.
- Symmetric matrixEigenvalues and eigenvectors
- A matrix equal to its own transpose.
T
- Tangent lineDifferentiation
- The straight line that touches a curve at one point and has the same slope there.
- Taylor seriesSequences and series
- A function written as an infinite sum of powers of x, with coefficients f^(n)(0) / n!.
- Trace and determinant checkEigenvalues and eigenvectors
- The eigenvalues add up to the trace (sum of diagonal entries) and multiply to the determinant.
- Trigonometric integralsIntegration techniques and uses
- Use identities to turn powers of sine and cosine into forms you can integrate.
- Types of discontinuityLimits and continuity
- A removable discontinuity is a single hole.
V
- VectorVectors, matrices and linear systems
- A quantity with a size and a direction, written as a list of numbers such as (3, 4).
- Vector spaceDeterminants, inverses and vector spaces
- A set of vectors you can add together and scale by numbers, while always staying inside the set.
- Volume of revolutionIntegration techniques and uses
- Spin a region around the x-axis.
W
- Work as an integralIntegration techniques and uses
- When a force changes with position, work is the integral of force over distance.