Multivariable calculus extends slope and area into higher dimensions
Multivariable—or multivariate—calculus extends one-variable differentiation and integration to functions of several inputs. Gradients, partials, multiple integrals, and vector fields become the toolkit for space-dependent change. Because a point can now be approached along infinitely many paths, limits and continuity behave in ways that surprise newcomers.
Multivariable calculus extends differentiation and integration from functions of one variable to functions of several, which makes it an elementary part of calculus on Euclidean space; its three-dimensional special case is often called vector calculus. Two differences from the one-dimensional case demand care. A point in higher dimensions can be approached along infinitely many routes rather than just from the left or right, and there are more kinds of geometric object: a function of two variables is a surface in three dimensions, while curves can also live in that space.
The first difference changes how limits and continuity are defined. Directional limits and derivatives work along a one-dimensional parametrised curve, reducing the problem to familiar single-variable terms, and a limit along a path depends on the route chosen, not only on the destination point. A general limit exists only when the limits along every possible path agree, and multivariate continuity is built from the same path idea.
The subject throws up counterintuitive results. A textbook function on the unit square, defined piecewise as y/x − y below the diagonal, x/y − x above it, 1 − x on the diagonal and zero elsewhere, is continuous in each variable separately yet fails to be continuous as a function of both together. Being continuous in each argument is therefore not enough.
The second difference produces several kinds of integration, line integrals, surface integrals and volume integrals, and because they are not unique, no single antiderivative or indefinite integral can be properly defined. Linearity and superposition, by contrast, carry over unchanged from single-variable calculus.
Source: Multivariable calculus
