Essential Mathematical Methods for the Physical Sciences

In Section A.9 of Appendix A we review the algebra of vectors, and in Chapter 1 we considered how to transform one vector into another using a linear operator. In this chapter and the next we discuss the calculus of vectors, i.e. the differentiation and integration both of vectors describing particular bodies, such as the velocity of a particle, and of vector fields, in which a vector is defined as a function of the coordinates throughout some volume (one-, two- or three-dimensional). Since the aim of this chapter is to develop methods for handling multi-dimensional physical situations, we will assume throughout that the functions with which we have to deal have sufficiently amenable mathematical properties, in particular that they are continuous and differentiable.

Differentiation of vectors

Let us consider a vector a that is a function of a scalar variable u. By this we mean that with each value of u we associate a vector a(u). For example, in Cartesian coordinates a(u) = ax(u)i + ay(u)j + az(u)k, where ax(u), ay(u) and az(u) are scalar functions of u and are the components of the vector a(u) in the x-, y- and z-directions respectively. We note that if a(u) is continuous at some point u = u0 then this implies that each of the Cartesian components ax(u), ay(u) and az(u) is also continuous there.