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This book introduces the concepts of tensor algebras and differentiable manifolds to the intermediate-level student. It describes analytical and geometrical structures built on these basic concepts. Those structures -- which include differential forms and their integration, flows, Lie derivatives, distributions and their integrability conditions, connections, and pseudo-Riemannian and symplectic manifolds -- are then applied to the description of the fundamental ideas and Hamiltonian and Lagrangian mechanics, and special and general relativity. This book is designed to be accessible to the mathematics or physics student with a good standard undergraduate background, who is interested in obtaining a broader perspective of the rich interplay of mathematics and physics before deciding on a specialty.
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Robert H. Wasserman is at University of Michigan (Emeritus).Review:
"Considering the complexity of the material, the tone is friendly; one can almost hear Wasserman lecturing. The presentation is rigorous without being pedantic. Excellent exercises. Highly recommended." --Choice
"First, the emphasis is placed on mathematical structures and all chapters are built around them. . . Second, the text is highly interactive: problems appear all the time and problems are turned into exercises. Third, the author does not shy away from . . . sending the reader to very precise references, with page numbers. Finally, the book strikes a balance between background and advanced topics, easy and difficult material, classical and modern methods, and theory and application. . . . a valuable addition to the courses available in manifold theory and tensor analysis." --SIAM Review
"A nice and comprehensive introduction. . .. clearly written and self-contained and, in particular, the author has succeeded in combining mathematical rigor with a certain degree of informality in a satisfactory way. . . . this work will certainly be appreciated by a wide audience, and may be recommended especially to advanced physics and mathematics students and beginning researchers." --Mathematical Reviews
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