## Integrating fields

This chapter shows how line integrals, surface integrals and volume integrals are defined as limits of sums and how they are evaluated. It illustrates their relevance to physical problems. The chapter introduces two important vector integral theorems known as Stokes’s theorem and Gauss’s theorem and indicates informally how they are derived. The basic strategy for evaluating line integrals, surface integrals and volume integrals is to express them in terms of ordinary definite integrals. A line integral is a generalisation of the ordinary definite integral to two and three dimensions, the path of integration being along any curve in space. Some line integrals have scalar values while others are vectors. One way in which scalar line integrals arise is in the description of energy exchange between a force field and a particle. A line integral is evaluated by introducing a coordinate system and expressing the line integral as one or more ordinary definite integrals.