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OverviewThe present volume supersedes my Introduction to Differentiable Manifolds written a few years back. I have expanded the book considerably, including things like the Lie derivative, and especially the basic integration theory of differential forms, with Stokes' theorem and its various special formulations in different contexts. The foreword which I wrote in the earlier book is still quite valid and needs only slight extension here. Between advanced calculus and the three great differential theories (differential topology, differential geometry, ordinary differential equations), there lies a no-man's-land for which there exists no systematic exposition in the literature. It is the purpose of this book to fill the gap. The three differential theories are by no means independent of each other, but proceed according to their own flavor. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them (immersions, embeddings, isomorphisms, etc.). One may also use differentiable structures on topological manifolds to determine the topological structure of the manifold (e.g. it la Smale [26]). Full Product DetailsAuthor: Serge LangPublisher: Springer-Verlag New York Inc. Imprint: Springer-Verlag New York Inc. Edition: 2nd ed. 1985. 2nd printing Dimensions: Width: 15.50cm , Height: 1.30cm , Length: 23.50cm Weight: 0.454kg ISBN: 9780387961132ISBN 10: 0387961135 Pages: 230 Publication Date: 03 October 1988 Audience: Professional and scholarly , Professional & Vocational Replaced By: 9780387943381 Format: Paperback Publisher's Status: Active Availability: Temporarily unavailable  The supplier advises that this item is temporarily unavailable. It will be ordered for you and placed on backorder. Once it does come back in stock, we will ship it out to you. Table of ContentsI Differential Calculus.- 1. Categories.- 2. Topological vector spaces.- 3. Derivatives and composition of map.- 4. Integration and Taylor’s formula.- 5. The inverse mapping theorem.- II Manifolds.- 1. Atlases, charts, morphisms.- 2. Submanifolds, immersions, submersions.- 3. Partitions of unity.- 4. Manifolds with boundary.- III Vector Bundles.- 1. Definition, pull-backs.- 2. The tangent bundle.- 3. Exact sequences of bundles.- 4. Operations on vector bundles.- 5. Splitting of vector bundles.- IV Vector Fields and Differential Equations.- 1. Existence theorem for differential equations.- 2. Vector fields, curves, and flows.- 3. Sprays.- 4. The exponential map.- 5. Existence of tubular neighborhoods.- 6. Uniqueness of tubular neighborhoods.- V Differential Forms.- 1. Vector fields, differential operators, brackets.- 2. Lie derivative.- 3. Exterior derivative.- 4. The canonical 2-form.- 5. The Poincaré lemma.- 6. Contractions and Lie derivative.- 7. Darboux theorem.- VI The Theorem of Frobenius.- 1. Statement of the theorem.- 2. Differential equations depending on a parameter.- 3. Proof of the theorem.- 4. The global formulation.- 5. Lie groups and subgroups.- VII Riemannian Metrics.- 1. Definition and functoriality.- 2. The Hilbert group.- 3. Reduction to the Hilbert group.- 4. Hilbertian tubular neighborhoods.- 5. Non-singular bilinear tensors.- 6. Riemannian metrics and sprays.- 7. The Morse-Palais lemma.- VIII Integration of Differential Forms.- 1. Sets of measure 0.- 2. Change of variables formula.- 3. Orientation.- 4. The measure associated with a differential form.- IX Stokes’ Theorem.- 1. Stokes’ theorem for a rectangular simplex.- 2. Stokes’ theorem on a manifold.- 3. Stokes’ theorem with singularities.- 4. The divergence theorem.- 5. Cauchy’stheorem.- 6. The residue theorem.- The Spectral Theorem.- 1 Hilbert space.- 2 Functionals and operators.- 3 Hermitian operators.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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