Mathematics for Physics: A Guided Tour for Graduate Students
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.
The second part (Chapters 10–14) focuses on modern differential 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 fields. The language of homology and cohomology is introduced and is used to investigate the influence of the global topology of a manifold on the fields that live in it and on the solutions of differential equations that constrain these fields.
Chapters 15 and 16 introduce the theory of group representations and their applications to quantum mechanics. Both finite groups and Lie groups are explored.
The last part (Chapters 17–19) 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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