Multivariable calculus, also known as multivariable analysis, is a branch of calculus that extends the concepts of single-variable calculus to functions of multiple variables. While single-variable calculus focuses on functions of one variable, such as \(f(x)\), multivariable calculus deals with functions of two or more variables, such as \(f(x, y)\) or \(g(x, y, z)\).
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Multivariable calculus studies derivatives and integrals of functions with several independent variables. A temperature field , for example, has partial derivatives describing coordinate-wise changes, while its gradient gives the direction and rate of steepest increase. The Fréchet derivative combines directional changes into one linear map; its coordinate representation is the Jacobian matrix.
For , the gradient is and the directional derivative along a unit vector is . The Hessian matrix describes second-order variation and, under the hypotheses of the second-derivative test, distinguishes a local minimum from a saddle point. Constrained variation leads to Lagrange multipliers and the implicit function theorem.
Multiple integrals accumulate quantities over regions, as when integrating a mass density to obtain total mass. A change of variables formula uses the absolute Jacobian determinant to account for stretching of volume. For vector fields, the fundamental theorem for line integrals, Stokes theorem and divergence theorem relate interior derivatives to boundary measurements. They extend the endpoint difference of the ordinary fundamental theorem of calculus to circulation and flux.