**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 nd 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 rst, and longest, part (Chapters 1 to 9, and the rst**

**semester in the classroom) to traditional mathematical methods. We explore**

**the analogy between linear operators acting on function spaces and matrices**

**acting on nite dimensional spaces, and use the operator language to provide**

**a uni ed framework for working with ordinary di erential 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 di erential 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.**

**The second part (Chapters 10 to 14) focuses on modern di erential geometry and topology, with an eye to its application to physics. The tools of**

**calculus on manifolds, especially the exterior calculus, are introduced, and used to investigate classical mechanics, electromagnetism, and non-abelian**

**gauge elds. The language of homology and cohomology is introduced and**

**is used to investigate the in uence of the global topology of a manifold on**

**the elds that live in it and on the solutions of di erential equations that**

**constrain these elds.**

**Chapters 15 and 16 introduce the theory of group representations and**

**their applications to quantum mechanics. Both nite groups and Lie groups**

**are explored.**

**The last part (Chapters 17 to 19) explores the theory of complex variables**

**and its applications. Although much of the material is standard, we make use**

**of the exterior calculus, and discuss rather more of the topological aspects of**

**analytic functions than is customary.**

**A cursory reading of the Contents of the book will show that there is**

**more material here than can be comfortably covered in two semesters. When**

**using the book as the basis for lectures in the classroom, we have found it**

**useful to tailor the presented material to the interests of our students.**

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