Differential Geometry with Applications to Mechanics and Physics  book cover
1st Edition

Differential Geometry with Applications to Mechanics and Physics

ISBN 9780824703851
Published September 12, 2000 by CRC Press
480 Pages

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Book Description

An introduction to differential geometry with applications to mechanics and physics. It covers topology and differential calculus in banach spaces; differentiable manifold and mapping submanifolds; tangent vector space; tangent bundle, vector field on manifold, Lie algebra structure, and one-parameter group of diffeomorphisms; exterior differential forms; Lie derivative and Lie algebra; n-form integration on n-manifold; Riemann geometry; and more. It includes 133 solved exercises.

Table of Contents

Part 1 Topology and differential calculus requirements: topology; differential calculus in Banach spaces; exercises. Part 2 Manifolds: introduction; differential manifolds; differential mappings; submanifolds; exercises. Part 3 Tangent vector space: tangent vector; tangent space; differential at a point; exercises. Part 4 Tangent bundle-vector field-one-parameter group lie algebra: introduction; tangent bundle; vector field on manifold; lie algebra structure; one-parameter group of diffeomorphisms; exercises. Part 5 Cotangent bundle-vector bundle of tensors: cotangent bundle and covector field; tensor algebra; exercises. Part 6 Exterior differential forms: exterior form at a point; differential forms on a manifold; pull-back of a differential form; exterior differentiation; orientable manifolds; exercises. Part 7 Lie derivative-lie group: lie derivative; inner product and lie derivative; Frobenius theorem; exterior differential systems; invariance of tensor fields; lie group and algebra; exercises. Part 8 Integration of forms: n-form integration on n-manifold; integral over a chain; Stokes' theorem; an introduction to cohomology theory; integral invariants; exercises. Part 9 Riemann geometry: Riemannian manifolds.

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"The book is written in a very understandable and systematic way, with a lot of figures. A very good feature of the book is a collection of more than 130 exercises and problems … . The book can be recommended for a wide range of students as a first book to read on the subject. It can be also useful for the preparation of courses on the topic."
- EMS Newsletter
". . .a self-contained introduction to differential geometry. . ..very succinct."
---Monatshefte für Mathematik