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.