Groups usually appear in physics as symmetries of the system or model we are studying. Often the symmetry operation involves a linear transformation, and this naturally leads to the idea of finding sets of matrices having the same multiplication table as the group. These sets are called representations of the group. Given a group, we endeavour to find and classify all possible representations.

Basic ideas

We begin with a rapid review of basic group theory.

Group axioms

A group G is a set with a binary operation that assigns to each ordered pair (g1, g2) of elements a third element, g3, usually written with multiplicative notation as g3 = g1g2. The binary operation, or product, obeys the following rules:

1. (i) Associativity: g1(g2g3) = (g1g2)g3.

2. (ii) Existence of an identity: there is an element e ∈ G such that eg = g for all g ∈ G.

3. (iii) Existence of an inverse: for each g ∈ G there is an element g–1 such that g–1g = e.

From these axioms there follow some conclusions that are so basic that they are often included in the axioms themselves, but since they are not independent, we state them as corollaries.

We begin our tour of useful mathematics with what is called the calculus of variations. Many physics problems can be formulated in the language of this calculus, and once they are there are useful tools to hand. In the text and associated exercises we will meet some of the equations whose solution will occupy us for much of our journey.

What is it good for?

The classical problems that motivated the creators of the calculus of variations include:

1. (i) Dido's problem: In Virgil's Aeneid, Queen Dido of Carthage must find the largest area that can be enclosed by a curve (a strip of bull's hide) of fixed length.

2. (ii) Plateau's problem: Find the surface of minimum area for a given set of bounding curves. A soap film on a wire frame will adopt this minimal-area configuration.

3. (iii) Johann Bernoulli's brachistochrone: A bead slides down a curve with fixed ends. Assuming that the total energy ½ mv2 + V(x) is constant, find the curve that gives the most rapid descent.

4. (iv) Catenary: Find the form of a hanging heavy chain of fixed length by minimizing its potential energy.

These problems all involve finding maxima or minima, and hence equating some sort of derivative to zero. In the next section we define this derivative, and show how to compute it.

Functionals

In variational problems we are provided with an expression J[y] that “eats” whole functions y(x) and returns a single number. Such objects are called functionals to distinguish them from ordinary functions.

Topology is the study of the consequences of continuity. We all know that a continuous real function defined on a connected interval and positive at one point and negative at another must take the value zero at some point between. This fact seems obvious – although a course of real analysis will convince you of the need for a proof. A less obvious fact, but one that follows from the previous one, is that a continuous function defined on the unit circle must posses two diametrically opposite points at which it takes the same value. To see that this is so, consider f (θ + π) – f (θ). This difference (if not initially zero, in which case there is nothing further to prove) changes sign as θ is advanced through π, because the two terms exchange roles. It was therefore zero somewhere. This observation has practical application in daily life: our local coffee shop contains four-legged tables that wobble because the floor is not level. They are round tables, however, and because they possess no misguided levelling screws all four legs have the same length. We are therefore guaranteed that by rotating the table about its centre through an angle of less than π/2 we will find a stable location. A ninety-degree rotation interchanges the pair of legs that are both on the ground with the pair that are rocking, and at the change-over point all four legs must be simultaneously on the ground.

Similar effects with a practical significance for physics appear when we try to extend our vector and tensor calculus from a local region to an entire manifold.

This book is based on a two-semester sequence of courses taught to incoming graduate students at the University of Illinois at Urbana-Champaign, primarily physics students but also some from other branches of the physical sciences. The courses aim to introduce students to some of the mathematical methods and concepts that they will find useful in their research. We have sought to enliven the material by integrating the mathematics with its applications. We therefore provide illustrative examples and problems drawn from physics. Some of these illustrations are classical but many are small parts of contemporary research papers. In the text and at the end of each chapter we provide a collection of exercises and problems suitable for homework assignments. The former are straightforward applications of material presented in the text; the latter are intended to be interesting, and take rather more thought and time.

We devote the first, and longest, part (Chapters 1–9, and the first semester in the classroom) to traditional mathematical methods. We explore the analogy between linear operators acting on function spaces and matrices acting on finite-dimensional spaces, and use the operator language to provide a unified framework for working with ordinary differential equations, partial differential equations and integral equations. The mathematical prerequisites are a sound grasp of undergraduate calculus (including the vector calculus needed for electricity and magnetism courses), elementary linear algebra and competence at complex arithmetic. Fourier sums and integrals, as well as basic ordinary differential equation theory, receive a quick review, but it would help if the reader had some prior experience to build on. Contour integration is not required for this part of the book.

In this section we will apply what we have learned about vectors and tensors in linear algebra to vector and tensor fields in a general curvilinear coordinate system. Our aim is to introduce the reader to the modern language of advanced calculus, and in particular to the calculus of differential forms on surfaces and manifolds.

Vector and covector fields

Vector fields – electric, magnetic, velocity fields, and so on – appear everywhere in physics. After perhaps struggling with it in introductory courses, we rather take the field concept for granted. There remain subtleties, however. Consider an electric field. It makes sense to add two field vectors at a single point, but there is no physical meaning to the sum of field vectors E(x1) and E(x2) at two distinct points. We should therefore regard all possible electric fields at a single point as living in a vector space, but each different point in space comes with its own field-vector space.

This view seems even more reasonable when we consider velocity vectors describing motion on a curved surface. A velocity vector lives in the tangent space to the surface at each point, and each of these spaces is a differently oriented subspace of the higherdimensional ambient space (see Figure 11.1).

Mathematicians call such a collection of vector spaces – one for each of the points in a surface – a vector bundle over the surface. Thus, the tangent bundle over a surface is the totality of all vector spaces tangent to the surface. Why a bundle? This word is used because the individual tangent spaces are not completely independent, but are tied together in a rather non-obvious way.