Glossary
Short, original definitions for higher-math vocabulary. Jump via the list or scroll. Terms are linked from topics and explainers across the site.
Basis
A linearly independent set of vectors that spans a vector space; every vector has unique coordinates relative to a basis. The Gram Schmidt calculator builds an orthogonal basis from a list of vectors. See linear algebra.
Bayes’ theorem
A rule relating P(H|E) to P(E|H), the prior P(H), and P(E). Used to update probabilities after observing evidence. See Bayes theorem without the fog. Compute a posterior on the Bayes theorem calculator.
Comparison test
A test for series of nonnegative terms. If 0 ≤ a_n ≤ b_n and Σ b_n converges, then Σ a_n converges. If a_n ≥ c_n ≥ 0 and Σ c_n diverges, then Σ a_n diverges. See convergence tests for series.
Conditional probability
P(A|B) is the probability of A given that B occurred; probability reweighted to the subspace B. See probability and statistics. See Bayes theorem without the fog. The Bayes theorem calculator computes P(H|E) from the other direction.
Constraint
An equation g(x) = c that restricts the points you may use when optimizing a function. Lagrange multipliers find candidates where the gradient of f is parallel to the gradient of g. See Lagrange multipliers intuition.
Continuity
A function is continuous at a if the limit as x→a equals f(a). Informally, the graph does not jump at a. See epsilon delta intuition.
Convergence (sequence/series)
A sequence converges if terms approach a limit. A series converges if its sequence of partial sums converges. See sequences, series, and infinite sums.
Curl
A local measure of rotation in a vector field. In the plane it is the scalar Qx minus Py. Positive means a counterclockwise tendency. In space it is a vector. Its direction is the axis of that rotation, and its length is the strength. A gradient of a smooth scalar function has curl zero. See Stokes theorem intuition. Estimate it on the divergence and curl calculator.
Definite integral
The signed accumulation of f from a to b, written ∫_a^b f(x) dx; equal to net signed area when f is integrable. See Riemann sum calculator.
Derivative
The instantaneous rate of change of f at a point; slope of the tangent / best linear approximation when it exists. See what a derivative measures.
Determinant
A scalar associated to a square matrix measuring oriented volume scaling; zero iff the matrix is singular. See 2 by 2 matrix calculator.
Differential equation
An equation relating a function to its derivatives. Solutions are functions (or families of functions), not just numbers. See differential equations. For y' + P(x) y = Q(x), the integrating factor calculator builds μ(x) and writes the steps. For a y'' + b y' + c y = 0, the second order differential equation calculator solves the characteristic equation and writes the steps. The Wronskian calculator tests linear independence of two or three candidate solutions. The Laplace transform calculator applies a short table and inverts a proper rational transform.
Divergence
A local measure of spreading in a vector field: the sum of the matching partial derivatives Px + Qy + Rz. Positive means a source tendency. Negative means a sink tendency. The divergence theorem equates the flux through a closed surface with the triple integral of this quantity. See multivariable and vector calculus. Estimate it on the divergence and curl calculator.
Eigenspace
For eigenvalue λ, the subspace of all eigenvectors for λ together with the zero vector; null space of A−λI. See eigenvalues in plain language.
Eigenvalue
A scalar λ such that A v = λ v for some nonzero vector v. Geometrically, a stretch factor along an invariant line. See eigenvalues in plain language.
Eigenvector
A nonzero vector sent to a scalar multiple of itself by a linear map or matrix. See eigenvalues in plain language.
Epsilon-delta definition
The precise definition of a limit: for every ε>0 there exists δ>0 such that 0<|x−a|<δ implies |f(x)−L|<ε. See epsilon delta intuition. Search for a sampled delta on the epsilon delta finder.
Expected value
The probability-weighted average of a random variable; center of mass of its distribution. See probability and statistics.
Gradient
The vector of first partial derivatives of a scalar function; points toward steepest ascent. See multivariable and vector calculus. Estimate partials and a directional derivative on the gradient and directional derivative calculator.
Implicit differentiation
Finding dy/dx from an equation F(x, y) = 0 by differentiating both sides with respect to x and treating y as a function of x. See implicit differentiation intuition.
Initial value problem (IVP)
A differential equation plus enough initial conditions to select a particular solution. See separable differential equations. A first order equation needs one condition, y(x0) = y0. The integrating factor calculator uses that condition when the equation is y' + P(x) y = Q(x). A second order equation needs two conditions, usually y and y' at one point. The second order calculator solves that system when the coefficients are constant. Once the transform of that problem is a proper rational function, the Laplace transform calculator inverts it.
Integral
An antiderivative (indefinite) or a definite accumulation (definite). Linked to derivatives by the Fundamental Theorem. See Riemann sum calculator.
Jacobian
The matrix of first partial derivatives of a map. It is the best linear approximation to that map at a point. The determinant, when the map is square, scales local volume. See Jacobian matrices in plain language and the Jacobian matrix calculator.
Lagrange multiplier
A number λ such that ∇f = λ ∇g at a constrained local max or min of f on g = c, when ∇g is nonzero. It measures how f responds to a small change in the allowed level of g. See Lagrange multipliers intuition.
Limit
The value a function or sequence approaches as the input approaches a point or infinity, when that approach settles. See epsilon delta intuition. The epsilon delta finder searches for a numerical delta once you propose L and epsilon.
Linear independence
A set of vectors is linearly independent if the only linear combination equaling zero uses all-zero coefficients. See linear algebra. For functions, a nonzero Wronskian at one point proves independence on any interval that contains that point. A zero at one point does not prove dependence. Compute that determinant on the Wronskian calculator.
Linear map (linear transformation)
A function between vector spaces that preserves addition and scalar multiplication. See linear algebra.
Matrix
A rectangular array representing a linear map relative to chosen bases, or a system of linear equations. See 2 by 2 matrix calculator.
Partial derivative
Derivative with respect to one variable while holding the others fixed. See Jacobian matrices in plain language.
Partial fraction decomposition
A way to write a rational function as a polynomial plus fractions whose denominators are powers of linear factors or irreducible quadratics. Used for antiderivatives and telescoping series. Compute one on the partial fraction decomposition calculator.
Partial sum
The sum of the first N terms of a series; the sequence of partial sums decides convergence. See sequences, series, and infinite sums.
Probability
A number between 0 and 1 assigned to events under a model, satisfying nonnegativity and countable additivity (in the standard axioms). See probability and statistics.
Random variable
A numerical quantity determined by the outcome of a random experiment; described by a distribution. See probability and statistics.
Ratio test
For a series Σ a_n, form L = lim |a_{n+1}/a_n| when the limit exists. If L < 1 the series converges absolutely. If L > 1 it diverges. If L = 1 the test is silent. See convergence tests for series.
Root test
For a series Σ a_n, form ρ = lim ( |a_n| )^{1/n} when the limit exists. The conclusions match the ratio test with ρ in place of L. Useful when terms are already written as nth powers. See convergence tests for series.
Sequence
An ordered list of numbers a₁, a₂, a₃, … indexed by positive integers (or nonnegative integers). See sequences, series, and infinite sums.
Series
An infinite sum Σ a_n, defined as the limit of its partial sums when that limit exists. See sequences, series, and infinite sums.
Slope field
A plot of short segments showing the slope f(t,y) of an ODE y'=f(t,y) at sample points; guides qualitative solution shapes. See separable differential equations.
Span
The set of all linear combinations of a given collection of vectors. See linear algebra.
Stokes theorem
The circulation of a smooth vector field F around the oriented boundary of a surface S equals the flux of curl F through S. Green’s theorem is the flat special case. See Stokes theorem intuition.
Vector
An element of a vector space; informally an arrow that can be added and scaled. See linear algebra.
Vector field
An assignment of a vector to each point of a region, for example the velocity of a flow. See Stokes theorem intuition.
Wronskian
The determinant of a matrix whose columns are a list of functions and their derivatives. For solutions of one linear homogeneous differential equation, a nonzero value means the solutions are linearly independent, and an identically zero Wronskian means they are dependent. A zero at one point, for arbitrary functions, does not prove dependence. Compute one on the Wronskian calculator. See differential equations.