Multivariable calculus
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Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable.
[edit] Typical operations
Typical operations in multivariable calculus are partial differentiation and integration over planar regions or solid bodies. Partial differentiation describes the rate of change of a function with respect to one or more of its variables, and thus is essentially a single-variable calculus notion in disguise. The more serious concept of differentiation in multivariable calculus requires the tools of linear algebra: the derivative of a function of several variables is understood to be a certain linear transformation which varies from point to point in the domain of the function. Multivariable integrals can be used to calculate areas on a surface or volumes of complicated solids. The fundamental theorem of calculus extends to the setting of multivariable calculus, first as particular theorems in vector calculus attributed to Gauss, Green, and Stokes and then as the general Stokes theorem which applies to integration of differential forms on manifolds.
[edit] Applications and uses
Multivariable calculus can be applied to analyze deterministic systems with multiple degrees of freedom. Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the system dynamics.
Multivariable calculus is used in many fields of natural and social science and engineering to model and study high-dimensional systems that exhibit deterministic behavior. Non-deterministic, or stochastic systems can be studied using a different kind of mathematics, such as stochastic calculus.
