10

Vector calculus

In chapter 7 we discussed the algebra of vectors, and in chapter 8 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.

10.1 Differentiation of vectors

Letusconsideravectorathat is a function of a scalar variableu.Bythis

we mean that with each value ofuwe associate a vectora(u). For example, in

Cartesian coordinatesa(u)=ax(u)i+ay(u)j+az(u)k,whereax(u),ay(u)andaz(u)

are scalar functions ofuand are the components of the vectora(u)inthex-,y-

andz- directions respectively. We note that ifa(u) is continuous at some point

u=u 0 then this implies that each of the Cartesian componentsax(u),ay(u)and

az(u) is also continuous there.

Let us consider the derivative of the vector functiona(u)withrespecttou.

The derivative of a vector function is defined in a similar manner to the ordinary

derivative of a scalar functionf(x) given in chapter 2. The small change in

the vectora(u) resulting from a small change ∆uin the value ofuis given by

∆a=a(u+∆u)−a(u) (see figure 10.1). The derivative ofa(u) with respect touis

defined to be

da

du

= lim

∆u→ 0

a(u+∆u)−a(u)

∆u

, (10.1)