Multivariable Calculus and Differential Geometry / Gerard Walschap

Von:

Walschap, Gerard, 1954- [aut]

Resource type: Ressourcentyp: Buch (Online)Buch (Online)Sprache: Englisch Reihen: De Gruyter eBook-Paket Mathematik | De Gruyter TextbookVerlag: Berlin ; Boston : De Gruyter, 2015Copyright-Datum: ©2015Beschreibung: 1 Online-Ressource (IX, 355 S.)ISBN:

9783110369540

Schlagwörter:

Andere physische Formen: 9783110369496 | Erscheint auch als: Multivariable calculus and differential geometry. Druck-Ausgabe Berlin : de Gruyter, 2015. ix, 355 SeitenDDC-Klassifikation:

516.36
515 23
516.3/6

MSC: MSC: *53-01 | 53A04 | 53A05 | 53A07 | 26-01RVK: RVK: SK 370Local classification: Lokale Notation: math 4.30LOC-Klassifikation:

QA303.2
QA641

DOI: DOI: 10.1515/9783110369540Online-Ressourcen:

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Inhalte:

2.10 Partitions of unity2.11 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 coordinates4.6.2 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 variation6.4 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.

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.

Zusammenfassung: <!doctype html public ""-//w3c//dtd html 4.0 transitional//en""> <html><head> <meta content=""text/html; charset=iso-8859-1"" http-equiv=content-type> <meta name=generator content=""mshtml 8.00.6001.23487""></head> <body> <P>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.</P></body></html>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. Gerard Walschap, University of Oklahoma, Norman, OK, USAOriginal version: Originalfassung: 2015Other relationships: Weitere Beziehungen:

Print version: Multivariable Calculus and Differential Geometry

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2015

In English