Chapter 8 - Differential Calculus with Several Independent Variables

Abstract

A derivative of a function of several variables with respect to one independent variable is called a partial derivative. The other variables are treated as constants during the differentiation. There are some useful identities that allow manipulations of expressions containing partial derivatives. The differential of a function of several variables (an exact differential) has one term for each variable, consisting of a partial derivative times the differential of the independent variable. This differential form delivers the value of an infinitesimal change in the function produced by infinitesimal changes in the independent variables. Differential forms exist that are not the differentials of any function. Such a differential form is called an inexact differential. Relative maxima and minima of a function of several variables are found by solving simultaneously the equations obtained by setting all partial derivatives equal to zero. Constrained maxima and minima of a function of several variables can be found by the method of Lagrange multipliers. The gradient operator is a vector derivative operator that produces a vector when applied to a scalar function. The divergence operator is a vector derivative operator that produces a scalar when applied to a vector function. The curl operator is a vector derivative operator that produces a vector when applied to a vector function. The Laplacian operator is equivalent to the divergence of the gradient of a scalar function.