Multivariable Calculus and Differential Geometry

✍ Scribed by Gerard Walschap

• Publisher: De Gruyter
• Year: 2015
• Tongue: English
• Leaves: 369
• Series: De Gruyter graduate,
• Edition: 1
• Category: Library

✦ Synopsis

About this book
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 and physics.

The main focus is on manifolds in Euclidean space and the metric properties they inherit from it. Among the topics discussed are curvature and how it affects the shape of space, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.

A thorough introduction to differential geometry
Covers all important concepts and theorems
Many examples including applications to physics

Author / Editor information
Gerard Walschap, University of Oklahoma, Norman, OK, USA.

✦ Table of Contents

Contents
Preface|v
1 Euclidean Space|1
1.1 Vector spaces|1
1.2 Linear transformations|6
1.3 Determinants|12
1.4 Euclidean spaces|19
1.5 Subspaces of Euclidean space|25
1.6 Determinants as volume|27
1.7 Elementary topology of Euclidean spaces|30
1.8 Sequences|36
1.9 Limits and continuity|41
1.10 Exercises|48
2 Differentiation|57
2.1 The derivative|57
2.2 Basic properties of the derivative|62
2.3 Differentiation of integrals|67
2.4 Curves|69
2.5 The inverse and implicit function theorems|75
2.6 The spectral theorem and scalar products|81
2.7 Taylor polynomials and extreme values|89
2.8 Vector fields|94
2.9 Lie brackets|103
2.10 Partitions of unity|108
2.11 Exercises|110
3 Manifolds|117
3.1 Submanifolds of Euclidean space|117
3.2 Differentiable maps on manifolds|124
3.3 Vector fields on manifolds|129
3.4 Lie groups|137
3.5 The tangent bundle|141
3.6 Covariant differentiation|143
3.7 Geodesics|148
3.8 The second fundamental tensor|153
3.9 Curvature|156
3.10 Sectional curvature|160
3.11 Isometries|163
3.12 Exercises|168
viii | Contents
4 Integration on Euclidean space|177
4.1 The integral of a function over a box|177
4.2 Integrability and discontinuities|181
4.3 Fubini’s theorem|187
4.4 Sard’s theorem|195
4.5 The change of variables theorem|198
4.6 Cylindrical and spherical coordinates|202
4.6.1 Cylindrical coordinates|202
4.6.2 Spherical coordinates|206
4.7 Some applications|210
4.7.1 Mass|211
4.7.2 Center of mass|211
4.7.3 Moment of inertia|213
4.8 Exercises|214
5 Differential Forms|221
5.1 Tensors and tensor fields|221
5.2 Alternating tensors and forms|224
5.3 Differential forms|232
5.4 Integration on manifolds|236
5.5 Manifolds with boundary|240
5.6 Stokes’ theorem|243
5.7 Classical versions of Stokes’ theorem|246
5.7.1 An application: the polar planimeter|249
5.8 Closed forms and exact forms|252
5.9 Exercises|257
6 Manifolds as metric spaces|267
6.1 Extremal properties of geodesics|267
6.2 Jacobi fields|271
6.3 The length function of a variation|275
6.4 The index form of a geodesic|278
6.5 The distance function|283
6.6 The Hopf-Rinow theorem|285
6.7 Curvature comparison|289
6.8 Exercises|292
7 Hypersurfaces|301
7.1 Hypersurfaces and orientation|301
7.2 The Gauss map|304
7.3 Curvature of hypersurfaces|308
7.4 The fundamental theorem for hypersurfaces|313
Contents | ix
7.5 Curvature in local coordinates|316
7.6 Convexity and curvature|318
7.7 Ruled surfaces|320
7.8 Surfaces of revolution|323
7.9 Exercises|328
Appendix A|339
Appendix B|345
Index|351

✦ Subjects

Multivariable Calculus, Differential Geometry, Vector Analysis, Topology