Multivariable calculus

The branch of calculus that studies functions of several variables, including their derivatives, integrals, and applications to geometry, physics, and optimization.

Partial derivative: A derivative measuring how a function changes with respect to one variable while other variables are held fixed. It isolates change along one coordinate direction.
Euclidean space: A geometric space with distances and angles defined by the Euclidean metric. It is the usual setting for multivariable functions and regions.
Lagrange multipliers: A method for finding extrema of a function subject to equality constraints. It turns constrained optimization into equations involving gradients.
Ordinary differential equation: A differential equation involving an unknown function of one independent variable and its derivatives. It models change along one independent variable rather than across space.
Gradient: A vector of a scalar function’s partial derivatives, pointing toward its steepest local increase. It combines directional rates of change into one vector.
Vector: A mathematical object with magnitude and direction, represented by coordinates in a vector space. Vectors describe points, directions, and rates of change.
Stokes' theorem: A theorem equating a surface integral of curl with a line integral around its boundary. It converts between circulation and rotation across a surface.
Partial differential equation: A differential equation involving partial derivatives of an unknown function of several variables. It is a principal application of multivariable differentiation.
Directional derivative: The rate at which a function changes at a point in a specified direction. It extends partial derivatives beyond coordinate directions.
Matrix: A rectangular array of numbers used to represent linear transformations and systems of equations. Matrices express derivatives of vector-valued functions.

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