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Mathematics for Physics: A Guided Tour for Graduate Students

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 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.