Multivariable Calculus and Differential Geometry

by Gerard Walschap

Overview

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.

Product Details

ISBN-13:	9783110369496
Publisher:	De Gruyter
Publication date:	06/12/2015
Series:	De Gruyter Textbook
Pages:	365
Product dimensions:	6.60(w) x 9.40(h) x 0.90(d)
Age Range:	18 Years

About the Author

Gerard Walschap, University of Oklahoma, Norman, OK, USA.

Table of 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 Differentiate 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 Geodesies 148

3.3 The second fundamental tensor 153

3.9 Curvature 156

3.10 Sectional curvature 160

3.11 Isometries 163

3.12 Exercises 168

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 geodesies 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

7.5 Curvature in local coordinates 316

7.6 Convexityand 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