Table of contents

FiniteDimensional Vector Spaces

InfiniteDimensional Vector Spaces

Complex Analysis

Differential Equations

Operators on Hilbert Spaces

Green’s Functions

Groups and Their Representations

Tensors and Manifolds

Lie Groups and Their Applications

Fiber Bundles
About this book
Introduction
The goal of this book is to expose the reader to the indispensable role that mathematicsoften very abstractplays in modern physics. Starting with the notion of vector spaces, the first half of the book develops topics as diverse as algebras, classical orthogonal polynomials, Fourier analysis, complex analysis, differential and integral equations, operator theory, and multidimensional Green's functions. The second half of the book introduces groups, manifolds, Lie groups and their representations, Clifford algebras and their representations, and fiber bundles and their applications to differential geometry and gauge theories.
This second edition is a substantial revision of the first one with a complete rewriting of many chapters and the addition of new ones, including chapters on algebras, representation of Clifford algebras and spinors, fiber bundles, and gauge theories. The spirit of the first edition, namely the balance between rigor and physical application, has been maintained, as is the abundance of historical notes and worked out examples that demonstrate the "unreasonable effectiveness of mathematics" in modern physics.
Einstein has famously said, "The most incomprehensible thing about nature is that it is comprehensible." What he had in mind was reiterated in another one of his famous quotes concerning the question of how " ... mathematics, being after all a product of human thought, is so admirably appropriate to the objects of reality." It is a question that comes to everyone's mind when encountering the highly abstract mathematics required for a deep understanding of modern physics. It is the experience that Eugene Wigner so profoundly described as "the unreasonable effectiveness of mathematics in the natural sciences."
Bibliographic information
 DOI https://doi.org/10.1007/9783319011950
 Copyright Information Springer International Publishing Switzerland 2013
 Publisher Name Springer, Cham
 eBook Packages Physics and Astronomy
 Print ISBN 9783319011943
 Online ISBN 9783319011950
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