| Description |
1 online resource (366) |
| Series |
De Gruyter graduate |
|
De Gruyter graduate.
|
| 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 |
| Summary |
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 |
| Analysis |
Differential geometry |
|
Riemannian geometry |
| Bibliography |
Includes bibliographical references (page 349) and index |
| Notes |
Print version record |
| Subject |
Geometry, Differential.
|
|
Calculus.
|
|
calculus.
|
|
MATHEMATICS -- Geometry -- General.
|
|
Geometry, Differential
|
| Form |
Electronic book
|
| ISBN |
3110369540 |
|
9783110369540 |
|
9783110392791 |
|
3110392798 |
|
