Cover......Page 1
Title......Page 3
Classroom resource materials......Page 5
Contents......Page 9
Preface......Page 13
Note to Students......Page 19
1.1 Introduction......Page 23
1.2 Arclength Parametrization......Page 36
1.3 Frenet Formulas......Page 39
1.4 Non-Unit Speed Curves......Page 49
1.5 Some Implications of Curvature and Torsion......Page 56
1.6 Green's Theorem and the Isoperimetric Inequality......Page 60
1.7 The Geometry of Curves and Maple......Page 64
2.1 Introduction......Page 89
2.2 The Geometry of Surfaces......Page 99
2.3 The Linear Algebra of Surfaces......Page 108
2.4 Normal Curvature......Page 113
2.5 Surfaces and Maple......Page 118
3.1 Introduction......Page 129
3.2 Calculating Curvature......Page 133
3.3 Surfaces of Revolution......Page 141
3.4 A Formula for Gauss Curvature......Page 145
3.5 Some Effects of Curvature(s)......Page 149
3.6 Surfaces of Delaunay......Page 155
3.7 Elliptic Functions, Maple and Geometry......Page 159
3.8 Calculating Curvature with Maple......Page 171
4.1 Introduction......Page 183
4.2 First Notions in Minimal Surfaces......Page 186
4.3 Area Minimization......Page 192
4.4 Constant Mean Curvature......Page 195
4.5 Harmonic Functions......Page 201
4.6 Complex Variables......Page 204
4.7 Isothermal Coordinates......Page 206
4.8 The Weierstrass-Enneper Representations......Page 209
4.9 Maple and Minimal Surfaces......Page 219
5.1 Introduction......Page 231
5.2 The Geodesic Equations and the Clairaut Relation......Page 237
5.3 A Brief Digression on Completeness......Page 247
5.4 Surfaces not in M3......Page 248
5.5 Isometries and Conformal Maps......Page 257
5.6 Geodesies and Maple......Page 263
5.7 An Industrial Application......Page 284
6.1 Introduction......Page 297
6.2 The Covariant Derivative Revisited......Page 299
6.3 Parallel Vector Fields and Holonomy......Page 302
6.4 Foucault's Pendulum......Page 306
6.5 The Angle Excess Theorem......Page 308
6.6 The Gauss-Bonnet Theorem......Page 311
6.7 Applications of Gauss-Bonnet......Page 314
6.8 Geodesic Polar Coordinates......Page 319
6.9 Maple and Holonomy......Page 327
7.1 The Euler-Lagrange Equations......Page 333
7.2 Transversality and Natural Boundary Conditions......Page 340
7.3 The Basic Examples......Page 344
7.4 Higher-Order Problems......Page 349
7.5 The Weierstrass it-Function......Page 356
7.6 Problems with Constraints......Page 368
7.7 Further Applications to Geometry and Mechanics......Page 378
7.8 The Pontryagin Maximum Principle......Page 388
7.9 An Application to the Shape of a Balloon......Page 393
7.10 The Calculus of Variations and Maple......Page 402
8.2 Manifolds......Page 419
8.3 The Covariant Derivative......Page 423
8.4 Christoffel Symbols......Page 431
8.5 Curvatures......Page 438
8.6 The Charming Doubleness......Page 452
A.2 Examples in Chapter 2......Page 459
A.6 Examples in Chapter 6......Page 460
A.8 Examples in Chapter 8......Page 461
B Hints and Solutions to Selected Problems......Page 463
C Suggested Projects for Differential Geometry......Page 475
Bibliography......Page 477
Index......Page 483
About the Author......Page 491
Color pages......Page 492
Back cover......Page 496