12
Line, surface and volume integrals
In Chapter 11 we encountered continuously varying scalar and vector elds and discussed
the action of various differential operators on them. There is often a need to consider
not only these differential operations, bu

10
Matrices and vector spaces
In Chapter 9 we dened a vector as a geometrical object which has both a magnitude
and a direction and which may be thought of as an arrow xed in our familiar
three-dimensional space, a space which, if we need to, we dene by r

11
Vector calculus
In Chapter 9 we discussed the algebra of vectors and in Chapter 10 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

9
Vector algebra
This chapter introduces space vectors and their manipulation. Firstly we deal with the
description and algebra of vectors, then we consider how vectors may be used to
describe lines, planes and spheres, and nally we look at the practical

8
Multiple integrals
Just as functions of several variables may be differentiated with respect to two or more
of them, so may their integrals with respect to more than one variable be formed. The
formal denitions of such multiple integrals are extensions

7
Partial differentiation
In Chapters 3 and 4 we discussed functions f of only one variable x, which were usually
written f (x). Certain constants and parameters may also have appeared in the denition
of f , e.g. f (x) = ax + 2 contains the constant 2 and

3
Differential calculus
This and the next chapter are concerned with the formalism of probably the most widely
used mathematical technique in the physical sciences, namely the calculus. The current
chapter deals with the process of differentiation whilst

2
Preliminary algebra
It is normal practice when starting the mathematical investigation of a physical problem
to assign algebraic symbols to the quantity or quantities whose values are sought, either
numerically or as explicit algebraic expressions. For

6
Series and limits
Many examples exist in the physical sciences of situations where we are presented with a
sum of terms to evaluate. As just two examples, there may be the need to add together
the contributions from successive slits in a diffraction gra

5
Complex numbers and
hyperbolic functions
This chapter is concerned with the representation and manipulation of complex
numbers. Complex numbers pervade this book, underscoring their wide application in
the mathematics of the physical sciences. Some elem

1
Arithmetic and geometry
The rst two chapters of this book review the basic arithmetic, algebra and geometry of
which a working knowledge is presumed in the rest of the text; many students will have
at least some familiarity with much, if not all, of it.

4
Integral calculus
As indicated at the start of the previous chapter, the differential calculus and its
complement, the integral calculus, together form the most widely used tool for the
analysis of physical systems. The link that connects the two is tha