TWENTY - INTRODUCTION TO VECTOR ANALYSIS

Abstract

This chapter provides an overview of vector analysis. The notion of a vector field is central to the subject of vector analysis. The chapter presents a definition of vector field in R2 that states, “Let Ω be a region in R2, then f is a vector field in R2 if f assigns to every x in Ω a unique vector f(x) in R2.” This definition can be extended to R3. A vector field in R3 is a function of which domain and range are subsets of R3. Vector fields arise in a great number of physical applications. There are, for example, mechanical force fields, magnetic fields, electric fields, velocity fields, and direction fields. The chapter presents a general definition of the line integral in the plane and discusses a result that gives an important relationship between line integrals and double integrals. It discusses Green's theorem in the plane, which provides a relationship between a line integral over a closed curve and a double integral over a region enclosed by that curve. As in the plane, the computation of a line integral in space is very easy if the integral is independent of the path. The chapter discusses the notion of a surface integral. This will enable to extend Green's theorem to integrals over regions in space. It discusses the divergence and curl of a vector field in R3 and discusses how these are used in the statement of two very important theorems about surface integrals—Stokes' theorem and the divergence theorem. The chapter discusses how, under certain conditions, it is possible to convert from one set of coordinates to another in double and triple integrals.