Tensor Analysis on Manifolds (eBook)
288 Seiten
Dover Publications (Verlag)
978-0-486-13923-4 (ISBN)
"e;This is a first-rate book and deserves to be widely read."e; — American Mathematical MonthlyDespite its success as a mathematical tool in the general theory of relativity and its adaptability to a wide range of mathematical and physical problems, tensor analysis has always had a rather restricted level of use, with an emphasis on notation and the manipulation of indices. This book is an attempt to broaden this point of view at the stage where the student first encounters the subject. The authors have treated tensor analysis as a continuation of advanced calculus, striking just the right balance between the formal and abstract approaches to the subject.The material proceeds from the general to the special. An introductory chapter establishes notation and explains various topics in set theory and topology. Chapters 1 and 2 develop tensor analysis in its function-theoretical and algebraic aspects, respectively. The next two chapters take up vector analysis on manifolds and integration theory. In the last two chapters (5 and 6) several important special structures are studied, those in Chapter 6 illustrating how the previous material can be adapted to clarify the ideas of classical mechanics. The text as a whole offers numerous examples and problems.A student with a background of advanced calculus and elementary differential equation could readily undertake the study of this book. The more mature the reader is in terms of other mathematical knowledge and experience, the more he will learn from this presentation.
Chapter 0/Set Theory and Topology0.1. SET THEORY 0.1.1. Sets 0.1.2. Set Operations 0.1.3. Cartesian Products 0.1.4. Functions 0.1.5. Functions and Set Operations 0.1.6. Equivalence Relations0.2. TOPOLOGY 0.2.1. Topologies 0.2.2. Metric Spaces 0.2.3. Subspaces 0.2.4. Product Topologies 0.2.5. Hausdorff Spaces 0.2.6. Continuity 0.2.7. Connectedness 0.2.8. Compactness 0.2.9. Local Compactness 0.2.10. Separability 0.2.11 ParacompactnessChapter 1/Manifolds1.1. Definition of a Mainifold1.2. Examples of Manifolds1.3. Differentiable Maps1.4. Submanifolds1.5. Differentiable Maps1.6. Tangents1.7. Coordinate Vector Fields1.8. Differential of a MapChapter 2/Tensor Algebra2.1. Vector Spaces2.2. Linear Independence2.3. Summation Convention2.4. Subspaces2.5. Linear Functions2.6. Spaces of Linear Functions2.7. Dual Space2.8. Multilinear Functions2.9. Natural Pairing2.10. Tensor Spaces2.11. Algebra of Tensors2.12. Reinterpretations2.13. Transformation Laws2.14. Invariants2.15. Symmetric Tensors2.16. Symmetric Algebra2.17. Skew-Symmetric Tensors2.18. Exterior Algebra2.19. Determinants2.20. Bilinear Forms2.21. Quadratic Forms2.22. Hodge Duality2.23. Symplectic FormsChapter 3/Vector Analysis on Manifolds3.1. Vector Fields3.2. Tensor Fields3.3. Riemannian Metrics3.4. Integral Curves3.5. Flows3.6. Lie Derivatives3.7. Bracket3.8. Geometric Interpretation of Brackets3.9. Action of Maps3.10. Critical Point Theory3.11. First Order Partial Differential Equations3.12. Frobenius' TheoremAppendix to Chapter 33A. Tensor Bundles3B. Parallelizable Manifolds3C. OrientabilityChapter 4/Integration Theory4.1. Introduction4.2. Differential Forms4.3. Exterior Derivatives4.4. Interior Products4.5. Converse of the Poincaré Lemma4.6. Cubical Chains4.7. Integration on Euclidean Spaces4.8. Integration of Forms4.9. Strokes' Theorem4.10. Differential SystemsChapter 5/Riemannian and Semi-riemannian Manifolds5.1. Introduction5.2. Riemannian and Semi-riemannian Metrics5.3. "Lengeth, Angle, Distance, and Energy"5.4. Euclidean Space5.5. Variations and Rectangles5.6. Flat Spaces5.7. Affine connexions5.8 Parallel Translation5.9. Covariant Differentiation of Tensor Fields5.10. Curvature and Torsion Tensors5.11. Connexion of a Semi-riemannian Structure5.12. Geodesics5.13. Minimizing Properties of Geodesics5.14. Sectional CurvatureChapter 6/Physical Application6.1 Introduction6.2. Hamiltonian Manifolds6.3. Canonical Hamiltonian Structure on the Cotangent Bundle6.4. Geodesic Spray of a Semi-riemannian Manifold6.5. Phase Space6.6. State Space6.7. Contact Coordinates6.8. Contact ManifoldsBibliographyIndex
| Erscheint lt. Verlag | 26.4.2012 |
|---|---|
| Reihe/Serie | Dover Books on Mathematics |
| Sprache | englisch |
| Maße | 140 x 140 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Schlagworte | abstract concepts • accessible introductions • advanced math • Advanced Topics • advanced undergraduate • Algebraic • bartle • Black Hole • books on differential equations • books on differential forms • books on differential geometries • books on graph theories • books on hilbert spaces • books on pure mathematics • bostock • Buck • Bundle • Carroll • chartrand • Churchill • Cole • Compactness • Coordinate Systems • Covariant • CRC • crittenden • Curvature • Curves • deep understanding • Derivatives • differentiable • Differential Equations • differential forms • Differential Geometry • Dover • Edwards • Einstein • Euclidean Geometry • exactly 200 • Farlow • Fourier • Frames • Functional Analysis • geodesic • Geometric • geometrical • Graphs • graph theory • GRE • Greenberg • haberman • hamiltonian • Hardy • hilbert space • hole thermodynamics • introductory text • Invariant • Kay • Krantz • linear algebra • manifold • manuals • Mappings • Mathematical Analysis • mathematical background • mathematical perspective • Mathematical Physics • mathematical proofs • mathematical rigor • math majors • math student • math text • Matrix • Mendelson • metric • missing concepts • Notation • partial differential • Physicists • physics texts • Pinsky • Pinter • pure math • Pure Mathematics • Quantum effects • recalling • Schutz • schwarzschild • self study • Self-study • serious student • Spaces • Spacetime • Special relativity • Springer • Stoll • Suggestion • synge • Taylor • tensor calculus • Tensors • terse • Textbook • Theorems • theoretical physics • theory applied • theory background • theory stands • Tiles • Topological • Topology • Torsion • Trudeau • Variables • Vector • Wald |
| ISBN-10 | 0-486-13923-9 / 0486139239 |
| ISBN-13 | 978-0-486-13923-4 / 9780486139234 |
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