Mathematics for Physicists

11 Line and multiple integrals

In Chapter 7 we extended the discussion of differentiation given in Chapter 3 to functions of several variables. In this chapter we will extend the discussion of integration given in Chapter 4 in a similar way. We will begin by discussing functions of two variables, which we will usually take to be the Cartesian co-ordinates x, y, although they could equally well be, for example, a position and a time. The discussion will then be generalised to three or more variables and to other co-ordinate systems, especially polar co-ordinates in three dimensions. This will form the basis for important applications in vector analysis, which is an essential tool in understanding topics such as electromagnetic fields, fluid dynamics and potential theory, and which will be discussed extensively in Chapter 12.

11.1 Line integrals

In this section, we first introduce line integrals and their properties in two dimensions and then briefly indicate their extension to three dimensions, which is relatively straightforward. In both cases we will use Cartesian co-ordinates.

11.1.1 Line integrals in a plane

Suppose y = f(x) is a real single-valued monotonic continuous function of x defined in some interval x₁ < x < x₂, as represented by the curve C shown in Figure 11.1a. Then, if P(x, y) is a real single-valued continuous function of x and y for all points on the curve C, the integral

(11.1)

is called a line integral and the symbol C on the integration sign indicates ...