Multivariable Calculus and Differential Geometry.:
This text is a modern in-depth study of the subject that includes all the material needed from linear algebra. It then goes on to investigate topics in differential geometry, such as manifolds in Euclidean space, curvature, and the generalization of the fundamental theorem of calculus known as Stoke...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin/Boston, Germany :
De Gruyter,
2015.
©2015 |
| Schriftenreihe: | De Gruyter graduate.
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | This text is a modern in-depth study of the subject that includes all the material needed from linear algebra. It then goes on to investigate topics in differential geometry, such as manifolds in Euclidean space, curvature, and the generalization of the fundamental theorem of calculus known as Stokes' theorem. |
| Beschreibung: | 1 online resource (366) |
| Bibliographie: | Includes bibliographical references (page 349) and index. |
| ISBN: | 3110369540 9783110369540 9783110392791 3110392798 |
Internformat
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| 245 | 1 | 0 | |a Multivariable Calculus and Differential Geometry. |
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| 264 | 4 | |c ©2015 | |
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| 504 | |a Includes bibliographical references (page 349) and index. | ||
| 505 | 0 | |a Preface -- 1 Euclidean Space -- 1.1 Vector spaces -- 1.2 Linear transformations -- 1.3 Determinants -- 1.4 Euclidean spaces -- 1.5 Subspaces of Euclidean space -- 1.6 Determinants as volume -- 1.7 Elementary topology of Euclidean spaces -- 1.8 Sequences -- 1.9 Limits and continuity -- 1.10 Exercises -- 2 Differentiation -- 2.1 The derivative -- 2.2 Basic properties of the derivative -- 2.3 Differentiation of integrals -- 2.4 Curves -- 2.5 The inverse and implicit function theorems -- 2.6 The spectral theorem and scalar products -- 2.7 Taylor polynomials and extreme values -- 2.8 Vector fields -- 2.9 Lie brackets -- 2.10 Partitions of unity -- 2.11 Exercises -- 3 Manifolds -- 3.1 Submanifolds of Euclidean space -- 3.2 Differentiablemaps on manifolds -- 3.3 Vector fields on manifolds -- 3.4 Lie groups -- 3.5 The tangent bundle -- 3.6 Covariant differentiation -- 3.7 Geodesics -- 3.8 The second fundamental tensor -- 3.9 Curvature -- 3.10 Sectional curvature -- 3.11 Isometries -- 3.12 Exercises -- 4 Integration on Euclidean space -- 4.1 The integral of a function over a box -- 4.2 Integrability and discontinuities -- 4.3 Fubini's theorem -- 4.4 Sard's theorem -- 4.5 The change of variables theorem -- 4.6 Cylindrical and spherical coordinates -- 4.6.1 Cylindrical coordinates -- 4.6.2 Spherical coordinates -- 4.7 Some applications -- 4.7.1 Mass -- 4.7.2 Center ofmass -- 4.7.3 Moment of inertia -- 4.8 Exercises -- 5 Differential Forms -- 5.1 Tensors and tensor fields -- 5.2 Alternating tensors and forms -- 5.3 Differential forms -- 5.4 Integration on manifolds -- 5.5 Manifolds with boundary -- 5.6 Stokes' theorem -- 5.7 Classical versions of Stokes' theorem -- 5.7.1 An application: the polar planimeter -- 5.8 Closed forms and exact forms -- 5.9 Exercises -- 6 Manifolds as metric spaces. | |
| 505 | 8 | |a 6.1 Extremal properties of geodesics -- 6.2 Jacobi fields -- 6.3 The length function of a variation -- 6.4 The index formof a geodesic -- 6.5 The distance function -- 6.6 The Hopf-Rinow theorem -- 6.7 Curvature comparison -- 6.8 Exercises -- 7 Hypersurfaces -- 7.1 Hypersurfaces and orientation -- 7.2 The Gaussmap -- 7.3 Curvature of hypersurfaces -- 7.4 The fundamental theorem for hypersurfaces -- 7.5 Curvature in local coordinates -- 7.6 Convexity and curvature -- 7.7 Ruled surfaces -- 7.8 Surfaces of revolution -- 7.9 Exercises -- Appendix A -- Appendix B -- Index. | |
| 520 | |a This text is a modern in-depth study of the subject that includes all the material needed from linear algebra. It then goes on to investigate topics in differential geometry, such as manifolds in Euclidean space, curvature, and the generalization of the fundamental theorem of calculus known as Stokes' theorem. | ||
| 650 | 0 | |a Geometry, Differential. |0 http://id.loc.gov/authorities/subjects/sh85054146 | |
| 650 | 0 | |a Calculus. |0 http://id.loc.gov/authorities/subjects/sh85018802 | |
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| 650 | 7 | |a Geometry, Differential |2 fast | |
| 653 | |a Differential geometry. | ||
| 653 | |a Riemannian geometry. | ||
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| contents | Preface -- 1 Euclidean Space -- 1.1 Vector spaces -- 1.2 Linear transformations -- 1.3 Determinants -- 1.4 Euclidean spaces -- 1.5 Subspaces of Euclidean space -- 1.6 Determinants as volume -- 1.7 Elementary topology of Euclidean spaces -- 1.8 Sequences -- 1.9 Limits and continuity -- 1.10 Exercises -- 2 Differentiation -- 2.1 The derivative -- 2.2 Basic properties of the derivative -- 2.3 Differentiation of integrals -- 2.4 Curves -- 2.5 The inverse and implicit function theorems -- 2.6 The spectral theorem and scalar products -- 2.7 Taylor polynomials and extreme values -- 2.8 Vector fields -- 2.9 Lie brackets -- 2.10 Partitions of unity -- 2.11 Exercises -- 3 Manifolds -- 3.1 Submanifolds of Euclidean space -- 3.2 Differentiablemaps on manifolds -- 3.3 Vector fields on manifolds -- 3.4 Lie groups -- 3.5 The tangent bundle -- 3.6 Covariant differentiation -- 3.7 Geodesics -- 3.8 The second fundamental tensor -- 3.9 Curvature -- 3.10 Sectional curvature -- 3.11 Isometries -- 3.12 Exercises -- 4 Integration on Euclidean space -- 4.1 The integral of a function over a box -- 4.2 Integrability and discontinuities -- 4.3 Fubini's theorem -- 4.4 Sard's theorem -- 4.5 The change of variables theorem -- 4.6 Cylindrical and spherical coordinates -- 4.6.1 Cylindrical coordinates -- 4.6.2 Spherical coordinates -- 4.7 Some applications -- 4.7.1 Mass -- 4.7.2 Center ofmass -- 4.7.3 Moment of inertia -- 4.8 Exercises -- 5 Differential Forms -- 5.1 Tensors and tensor fields -- 5.2 Alternating tensors and forms -- 5.3 Differential forms -- 5.4 Integration on manifolds -- 5.5 Manifolds with boundary -- 5.6 Stokes' theorem -- 5.7 Classical versions of Stokes' theorem -- 5.7.1 An application: the polar planimeter -- 5.8 Closed forms and exact forms -- 5.9 Exercises -- 6 Manifolds as metric spaces. 6.1 Extremal properties of geodesics -- 6.2 Jacobi fields -- 6.3 The length function of a variation -- 6.4 The index formof a geodesic -- 6.5 The distance function -- 6.6 The Hopf-Rinow theorem -- 6.7 Curvature comparison -- 6.8 Exercises -- 7 Hypersurfaces -- 7.1 Hypersurfaces and orientation -- 7.2 The Gaussmap -- 7.3 Curvature of hypersurfaces -- 7.4 The fundamental theorem for hypersurfaces -- 7.5 Curvature in local coordinates -- 7.6 Convexity and curvature -- 7.7 Ruled surfaces -- 7.8 Surfaces of revolution -- 7.9 Exercises -- Appendix A -- Appendix B -- Index. |
| ctrlnum | (OCoLC)911847214 |
| dewey-full | 516.3/6 |
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| dewey-ones | 516 - Geometry |
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| discipline | Mathematik |
| format | Electronic eBook |
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| indexdate | 2024-11-27T13:26:40Z |
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| series | De Gruyter graduate. |
| series2 | De Gruyter graduate. |
| spelling | Walschap, Gerard. Multivariable Calculus and Differential Geometry. Berlin/Boston, Germany : De Gruyter, 2015. ©2015 1 online resource (366) text txt rdacontent computer c rdamedia online resource cr rdacarrier De Gruyter graduate. Print version record. Includes bibliographical references (page 349) and index. Preface -- 1 Euclidean Space -- 1.1 Vector spaces -- 1.2 Linear transformations -- 1.3 Determinants -- 1.4 Euclidean spaces -- 1.5 Subspaces of Euclidean space -- 1.6 Determinants as volume -- 1.7 Elementary topology of Euclidean spaces -- 1.8 Sequences -- 1.9 Limits and continuity -- 1.10 Exercises -- 2 Differentiation -- 2.1 The derivative -- 2.2 Basic properties of the derivative -- 2.3 Differentiation of integrals -- 2.4 Curves -- 2.5 The inverse and implicit function theorems -- 2.6 The spectral theorem and scalar products -- 2.7 Taylor polynomials and extreme values -- 2.8 Vector fields -- 2.9 Lie brackets -- 2.10 Partitions of unity -- 2.11 Exercises -- 3 Manifolds -- 3.1 Submanifolds of Euclidean space -- 3.2 Differentiablemaps on manifolds -- 3.3 Vector fields on manifolds -- 3.4 Lie groups -- 3.5 The tangent bundle -- 3.6 Covariant differentiation -- 3.7 Geodesics -- 3.8 The second fundamental tensor -- 3.9 Curvature -- 3.10 Sectional curvature -- 3.11 Isometries -- 3.12 Exercises -- 4 Integration on Euclidean space -- 4.1 The integral of a function over a box -- 4.2 Integrability and discontinuities -- 4.3 Fubini's theorem -- 4.4 Sard's theorem -- 4.5 The change of variables theorem -- 4.6 Cylindrical and spherical coordinates -- 4.6.1 Cylindrical coordinates -- 4.6.2 Spherical coordinates -- 4.7 Some applications -- 4.7.1 Mass -- 4.7.2 Center ofmass -- 4.7.3 Moment of inertia -- 4.8 Exercises -- 5 Differential Forms -- 5.1 Tensors and tensor fields -- 5.2 Alternating tensors and forms -- 5.3 Differential forms -- 5.4 Integration on manifolds -- 5.5 Manifolds with boundary -- 5.6 Stokes' theorem -- 5.7 Classical versions of Stokes' theorem -- 5.7.1 An application: the polar planimeter -- 5.8 Closed forms and exact forms -- 5.9 Exercises -- 6 Manifolds as metric spaces. 6.1 Extremal properties of geodesics -- 6.2 Jacobi fields -- 6.3 The length function of a variation -- 6.4 The index formof a geodesic -- 6.5 The distance function -- 6.6 The Hopf-Rinow theorem -- 6.7 Curvature comparison -- 6.8 Exercises -- 7 Hypersurfaces -- 7.1 Hypersurfaces and orientation -- 7.2 The Gaussmap -- 7.3 Curvature of hypersurfaces -- 7.4 The fundamental theorem for hypersurfaces -- 7.5 Curvature in local coordinates -- 7.6 Convexity and curvature -- 7.7 Ruled surfaces -- 7.8 Surfaces of revolution -- 7.9 Exercises -- Appendix A -- Appendix B -- Index. This text is a modern in-depth study of the subject that includes all the material needed from linear algebra. It then goes on to investigate topics in differential geometry, such as manifolds in Euclidean space, curvature, and the generalization of the fundamental theorem of calculus known as Stokes' theorem. Geometry, Differential. http://id.loc.gov/authorities/subjects/sh85054146 Calculus. http://id.loc.gov/authorities/subjects/sh85018802 Géométrie différentielle. Calcul infinitésimal. calculus. aat MATHEMATICS Geometry General. bisacsh Geometry, Differential fast Differential geometry. Riemannian geometry. Print version: Walschap, Gerard, 1954- Multivariable calculus and differential geometry. Berlin : De Gruyter, [2015] 3110369494 (DLC) 2015458315 (OCoLC)904420460 De Gruyter graduate. http://id.loc.gov/authorities/names/no2011117424 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=1017025 Volltext |
| spellingShingle | Walschap, Gerard Multivariable Calculus and Differential Geometry. De Gruyter graduate. Preface -- 1 Euclidean Space -- 1.1 Vector spaces -- 1.2 Linear transformations -- 1.3 Determinants -- 1.4 Euclidean spaces -- 1.5 Subspaces of Euclidean space -- 1.6 Determinants as volume -- 1.7 Elementary topology of Euclidean spaces -- 1.8 Sequences -- 1.9 Limits and continuity -- 1.10 Exercises -- 2 Differentiation -- 2.1 The derivative -- 2.2 Basic properties of the derivative -- 2.3 Differentiation of integrals -- 2.4 Curves -- 2.5 The inverse and implicit function theorems -- 2.6 The spectral theorem and scalar products -- 2.7 Taylor polynomials and extreme values -- 2.8 Vector fields -- 2.9 Lie brackets -- 2.10 Partitions of unity -- 2.11 Exercises -- 3 Manifolds -- 3.1 Submanifolds of Euclidean space -- 3.2 Differentiablemaps on manifolds -- 3.3 Vector fields on manifolds -- 3.4 Lie groups -- 3.5 The tangent bundle -- 3.6 Covariant differentiation -- 3.7 Geodesics -- 3.8 The second fundamental tensor -- 3.9 Curvature -- 3.10 Sectional curvature -- 3.11 Isometries -- 3.12 Exercises -- 4 Integration on Euclidean space -- 4.1 The integral of a function over a box -- 4.2 Integrability and discontinuities -- 4.3 Fubini's theorem -- 4.4 Sard's theorem -- 4.5 The change of variables theorem -- 4.6 Cylindrical and spherical coordinates -- 4.6.1 Cylindrical coordinates -- 4.6.2 Spherical coordinates -- 4.7 Some applications -- 4.7.1 Mass -- 4.7.2 Center ofmass -- 4.7.3 Moment of inertia -- 4.8 Exercises -- 5 Differential Forms -- 5.1 Tensors and tensor fields -- 5.2 Alternating tensors and forms -- 5.3 Differential forms -- 5.4 Integration on manifolds -- 5.5 Manifolds with boundary -- 5.6 Stokes' theorem -- 5.7 Classical versions of Stokes' theorem -- 5.7.1 An application: the polar planimeter -- 5.8 Closed forms and exact forms -- 5.9 Exercises -- 6 Manifolds as metric spaces. 6.1 Extremal properties of geodesics -- 6.2 Jacobi fields -- 6.3 The length function of a variation -- 6.4 The index formof a geodesic -- 6.5 The distance function -- 6.6 The Hopf-Rinow theorem -- 6.7 Curvature comparison -- 6.8 Exercises -- 7 Hypersurfaces -- 7.1 Hypersurfaces and orientation -- 7.2 The Gaussmap -- 7.3 Curvature of hypersurfaces -- 7.4 The fundamental theorem for hypersurfaces -- 7.5 Curvature in local coordinates -- 7.6 Convexity and curvature -- 7.7 Ruled surfaces -- 7.8 Surfaces of revolution -- 7.9 Exercises -- Appendix A -- Appendix B -- Index. Geometry, Differential. http://id.loc.gov/authorities/subjects/sh85054146 Calculus. http://id.loc.gov/authorities/subjects/sh85018802 Géométrie différentielle. Calcul infinitésimal. calculus. aat MATHEMATICS Geometry General. bisacsh Geometry, Differential fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85054146 http://id.loc.gov/authorities/subjects/sh85018802 |
| title | Multivariable Calculus and Differential Geometry. |
| title_auth | Multivariable Calculus and Differential Geometry. |
| title_exact_search | Multivariable Calculus and Differential Geometry. |
| title_full | Multivariable Calculus and Differential Geometry. |
| title_fullStr | Multivariable Calculus and Differential Geometry. |
| title_full_unstemmed | Multivariable Calculus and Differential Geometry. |
| title_short | Multivariable Calculus and Differential Geometry. |
| title_sort | multivariable calculus and differential geometry |
| topic | Geometry, Differential. http://id.loc.gov/authorities/subjects/sh85054146 Calculus. http://id.loc.gov/authorities/subjects/sh85018802 Géométrie différentielle. Calcul infinitésimal. calculus. aat MATHEMATICS Geometry General. bisacsh Geometry, Differential fast |
| topic_facet | Geometry, Differential. Calculus. Géométrie différentielle. Calcul infinitésimal. calculus. MATHEMATICS Geometry General. Geometry, Differential |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=1017025 |
| work_keys_str_mv | AT walschapgerard multivariablecalculusanddifferentialgeometry |