Multivariable Calculus and Differential Geometry

Name: Multivariable Calculus and Differential Geometry
Brand: De Gruyter
Price: 34.95 EUR
Availability: OnlineOnly

Gerard Walschap(Autor*in)

De Gruyter (Verlag)

1. Auflage

Erschienen am 1. Juli 2015

X, 355 Seiten

E-Book

PDF mit Wasserzeichen-DRM

Systemvoraussetzungen

978-3-11-036954-0 (ISBN)

34,95 €inkl. 7% MwSt.

Systemvoraussetzungen

für PDF mit Wasserzeichen-DRM

E-Book Einzellizenz

Als Download verfügbar

Beschreibung

Weitere Details

Weitere Ausgaben

Person

Inhalt

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
6.2 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

Systemvoraussetzungen

Als PDF speichern Als Link merken