7 - Calculus with several independent variables

Abstract

This chapter discusses the functions of more than one independent variable. Functions of several independent variables are described in the beginning of the chapter. These include changes in a function of several variables, partial derivatives, and differentials. The calculus of functions of several independent variables is a natural extension of the calculus of functions of one independent variable. The partial derivative is the first important quantity. Change of variables is then discussed as a part of the chapter. Additional useful information on partial derivatives is also provided in a segment. Exact and inexact differentials are described including integral factors. Line integrals, line integrals of exact differentials, line integrals of inexact differentials, line integrals with three integration variables, and line integrals in thermodynamics are also discussed in the chapter. Multiple integrals are detailed next, focusing on the double integral represented as a volume, multiple integrals in quantum mechanics, and changing variables in multiple integrals also form a part of the chapter. Vector derivative operators, which is another topic explained in the chapter, include information on vector derivatives in Cartesian coordinates, vector derivatives in other coordinate systems, and gradients in orthogonal coordinates. Maximum and minimum values of functions of several variables focus on constrained maximum/ minimum problems and Lagrange's method of undetermined multipliers are also detailed in the chapter.