Analysis on manifolds / James R. Munkres.
Series Advanced book classicsEditor: Boulder, Colo. : Westview Press, Advanced Book Program, c1991Descripción: xi, 366 p. : il. ; 24 cmISBN: 0201315963Tema(s): Mathematical analysis | Manifolds (Mathematics)Otra clasificación: 58-01 (26-01)PREFACE v CHAPTER [1] The Algebra and Topology of Rn [1] $1. Review of Linear Algebra [1] §2. Matrix Inversion and Determinants [11] §3. Review of Topology in Rn [25] §4. Compact Subspaces and Connected Subspaces of Rn [32] CHAPTER 2 Differentiation [41] §5- The Derivative [41] §6. Continuously Differentiable Functions [49] 57- The Chain Rule [56] §8. The Inverse Function Theorem [63] ‘§9. The Implicit Function Theorem [71] CHAPTER 3 Integration $10. The Integral over a Rectangle [81] $11. Existence of the Integral [91] $12. Evaluation of the Integral [98] $13. The Integral over a Bounded Set [104] $14. Rectifiable Sets [112] $15. Improper Integrals [121] CHAPTER 4 Change of Variables $16. Partitions of Unity [136] $17. The Change of Variables Theorem [144] $18. Diffeomorphisms in Rn [152] $19. Proof of the Change of Variables Theorem [161] $20- Applications of Change of Variables [169] CHAPTER 5 Manifolds [179] §21. The Volume of a Parallelopiped [180] §22. The Volume of a Parametrized-Manifold [188] §23. Manifolds in Rn [196] §24. The Boundary of a Manifold [203] §25. Integrating a Scalar Function over a Manifold [209] CHAPTER [6] Differential Forms [219] §26. Multilinear Algebra [220] §27. Alternating Tensors [226] §28. The Wedge Product [236] §29. Tangent Vectors and Differential Forms [244] §30. The Differential Operator [252] *§31. Application to Vector and Scalar Fields [262] §32. The Action of a Differentiable Map [267] CHAPTER 7 Stokes’ Theorem [275] $33. Integrating Forms over Parametrized-Manifolds [275] $34. Orientable Manifolds [281] $35. Integrating Forms over Oriented Manifolds [293] *$36. A Geometric Interpretation of Forms and Integrals [297] $37. The Generalized Stokes’ Theorem [301] *$38. Applications to Vector Analysis [310] CHAPTER 8 Closed Forms and Exact Forms [323] $39. The Poincare Lemma [324] $40. The deRham Groups of Punctured Euclidean Space [334] CHAPTER 9 Epilogue—Life Outside Rn [345] $41. Differentiable Manifolds and Riemannian Manifolds [345] BIBLIOGRAPHY [359] INDEX [361]
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 58 M966 (Browse shelf) | Available | A-8413 |
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Incluye referencias bibliográficas (p. 359-360) e índice.
PREFACE v --
CHAPTER [1] --
The Algebra and Topology of Rn [1] --
$1. Review of Linear Algebra [1] --
§2. Matrix Inversion and Determinants [11] --
§3. Review of Topology in Rn [25] --
§4. Compact Subspaces and Connected Subspaces of Rn [32] --
CHAPTER 2 Differentiation [41] --
§5- The Derivative [41] --
§6. Continuously Differentiable Functions [49] --
57- The Chain Rule [56] --
§8. The Inverse Function Theorem [63] --
Ԥ9. The Implicit Function Theorem [71] --
CHAPTER 3 Integration --
$10. The Integral over a Rectangle [81] --
$11. Existence of the Integral [91] --
$12. Evaluation of the Integral [98] --
$13. The Integral over a Bounded Set [104] --
$14. Rectifiable Sets [112] --
$15. Improper Integrals [121] --
CHAPTER 4 Change of Variables --
$16. Partitions of Unity [136] --
$17. The Change of Variables Theorem [144] --
$18. Diffeomorphisms in Rn [152] --
$19. Proof of the Change of Variables Theorem [161] --
$20- Applications of Change of Variables [169] --
CHAPTER 5 Manifolds [179] --
§21. The Volume of a Parallelopiped [180] --
§22. The Volume of a Parametrized-Manifold [188] --
§23. Manifolds in Rn [196] --
§24. The Boundary of a Manifold [203] --
§25. Integrating a Scalar Function over a Manifold [209] --
CHAPTER [6] --
Differential Forms [219] --
§26. Multilinear Algebra [220] --
§27. Alternating Tensors [226] --
§28. The Wedge Product [236] --
§29. Tangent Vectors and Differential Forms [244] --
§30. The Differential Operator [252] --
*§31. Application to Vector and Scalar Fields [262] --
§32. The Action of a Differentiable Map [267] --
CHAPTER 7 Stokes’ Theorem [275] --
$33. Integrating Forms over Parametrized-Manifolds [275] --
$34. Orientable Manifolds [281] --
$35. Integrating Forms over Oriented Manifolds [293] --
*$36. A Geometric Interpretation of Forms and Integrals [297] --
$37. The Generalized Stokes’ Theorem [301] --
*$38. Applications to Vector Analysis [310] --
CHAPTER 8 Closed Forms and Exact Forms [323] --
$39. The Poincare Lemma [324] --
$40. The deRham Groups of Punctured Euclidean Space [334] --
CHAPTER 9 Epilogue—Life Outside Rn [345] --
$41. Differentiable Manifolds and Riemannian Manifolds [345] --
BIBLIOGRAPHY [359] --
INDEX [361] --
MR, 92d:58001
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