Analysis & Calculus

An inverse function reverses a bijective function, assigning each output its unique original input.

The Jacobian matrix collects a function’s first-order partial derivatives and, when the function is differentiable, represents its local linear approximation.

Jensen’s inequality states that a convex function evaluated at an average cannot exceed the average of its values.

Karl Weierstrass was a German mathematician whose work on rigorous foundations, complex functions, and approximation helped shape modern mathematical analysis.

A second-order differential operator that measures local spatial variation and underlies equations of potential theory, diffusion, waves, and quantum mechanics.

Laplace’s equation is a linear partial differential equation describing harmonic functions and source-free equilibrium fields.

The Lebesgue integral defines integration through measure, extending classical integration and providing powerful theorems for convergence and probability.

Lebesgue measure extends length, area, and volume to a broad class of sets, providing a foundation for modern integration and probability.

Swiss mathematician and physicist whose discoveries and writings transformed analysis, number theory, geometry, and mechanics.

A limit describes the value approached by a function or sequence under a precisely specified process of convergence.

A line integral accumulates a scalar quantity or the tangential component of a vector field along a curve.

Lipschitz continuity bounds changes in a function’s output by a fixed multiple of changes in its input.

The logistic function is an S-shaped mathematical function used to describe bounded growth, transform log-odds into probabilities, and provide nonlinear activation in neural networks.

An Lp space is a space of measurable functions, identified up to equality almost everywhere, whose magnitude satisfies an integrability or essential boundedness condition.

Mathematical analysis studies limits, continuity, differentiation, integration, and the behavior of functions and infinite processes.

The mean value theorem states that a differentiable real function attains its average rate of change at some interior point of an interval.

A measurable function is a map whose inverse images of measurable sets are measurable, providing a foundation for integration and probability.

A measurable space is a set equipped with a sigma-algebra specifying which subsets are measurable, without assigning them a measure.

Measure theory studies generalized notions of size for sets and provides a rigorous foundation for integration and probability.

The method of exhaustion establishes exact areas and volumes through finite geometric approximations whose errors can be made arbitrarily small.

A metric completion embeds a metric space densely and isometrically into a complete metric space, supplying precisely its missing Cauchy limits.

A metric space is a set equipped with a distance function that satisfies positivity, symmetry, and the triangle inequality.

A function that consistently preserves or reverses the ordering of its inputs, with strict and non-strict variants.

A norm measures the size of a vector, generalizing length through definiteness, absolute homogeneity, and the triangle inequality.

A normed vector space is a real or complex vector space equipped with a norm that measures vector size and induces a compatible metric.

A null set is a measurable set assigned measure zero, expressing negligible size relative to a specified measure.

An open ball is the set of points whose distance from a specified center is strictly less than a positive radius.

A partial derivative measures how a multivariable function changes with one input while its other inputs remain fixed.

A partial differential equation relates a function of several independent variables to its partial derivatives, describing spatial variation, evolution, and other multivariable relationships.

Pointwise convergence is convergence of a sequence of functions at each fixed point of their common domain, without requiring uniform control across that domain.

Poisson's equation relates the Laplacian of an unknown function to a prescribed source, describing potentials and steady-state fields.

A power law relates quantities through a constant exponent, producing scale-invariant relationships and, in probability, distributions with slowly decaying tails.

A power series is an infinite sum of nonnegative integer powers of a variable, used to represent functions and encode sequences.

The Riemann integral defines accumulated quantities through limits of finite sums and exists for bounded functions whose discontinuities form a set of measure zero.

A Riemann sum approximates an integral by adding function values multiplied by the widths of corresponding subintervals.

The Riemann zeta function is a complex function whose analytic properties encode fundamental information about prime numbers and their distribution.

A saddle point is a stationary point with nearby values both above and below it, or a solution defined by opposing minimization and maximization conditions.

A self-adjoint operator equals its adjoint, including its domain, and admits a real spectral representation fundamental to analysis and quantum mechanics.

A Sobolev space is a complete function space that measures both integrability and generalized differentiability, providing a principal framework for partial differential equations and variational problems.

The softmax function converts real-valued scores into a normalized probability distribution by exponentiating each score and dividing by the sum of all exponentials.

Stokes’ theorem equates integration of a derivative over an oriented region with integration of the original quantity over its boundary.

A surjective function maps its domain onto its entire codomain, so every element of the codomain has at least one preimage.

A Taylor series is a power series whose coefficients are determined by a function’s derivatives at a chosen point, providing local representations and polynomial approximations.

Australian-American mathematician whose research spans harmonic analysis, partial differential equations, combinatorics, and number theory.

Uniform continuity requires a single input-distance bound to control output differences everywhere in a function’s domain.

Uniform convergence is convergence of functions in which one error bound eventually holds simultaneously at every point of the domain.

A vector field assigns a vector to each point of a space, describing quantities such as velocity, force, or local directions of motion.

A weak solution satisfies a differential equation through integral identities or distributional derivatives, allowing less regularity than a classical solution.