Multivariable Calculus and Differential Geometry

This book offers an introduction to differential geometry for the non-specialist. It includes most of the required material from multivariable calculus, linear algebra, and basic analysis. An intuitive approach and a minimum of prerequisites make it a valuable companion for students of mathematics a...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
1. Verfasser: Walschap, Gerard
Format: E-Book Buch
Sprache:Englisch
Veröffentlicht: Germany De Gruyter 2015
De Gruyter, Inc
Ausgabe:1
Schriftenreihe:De Gruyter Textbook
Schlagworte:
Algebras, Linear
Calculus
Differential geometry
Differentialgeometrie
Geometry
Geometry, Differential
Mathematical analysis
MATHEMATICS
MATHEMATICS / Geometry / Analytic
MATHEMATICS / Geometry / Differential
MATHEMATICS / Geometry / General
MATHEMATICS / Mathematical Analysis
Riemannian geometry
Riemannsche Geometrie
Riemannian geometry
Riemannsche Geometrie
Differentialgeometrie
Differential geometry
ISBN:3110369540, 9783110369540, 3110369494, 9783110369496
Online-Zugang:Volltext
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Inhaltsangabe:
  • 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
  • Intro -- 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. Manifolds as metric spaces
  • 3. Manifolds
  • 1. Euclidean Space
  • Index
  • 7. Hypersurfaces
  • 4. Integration on Euclidean space
  • -
  • /
  • Contents
  • Frontmatter --
  • 2. Differentiation
  • Preface
  • Appendix B
  • 5. Differential Forms
  • Appendix A