Elementary Differential Geometry, Revised 2nd Edition

Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces. The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard. This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs Mathematica or Maple, as well as additional computer exercises. As with the Second Edition, this material supplements the content but no computer skill is necessary to take full advantage of this comprehensive text. *Fortieth anniversary of publication! Over 36,000 copies sold worldwide *Accessible, practical yet rigorous approach to a complex topic--also suitable for self-study *Extensive update of appendices on Mathematica and Maple software packages *Thorough streamlining of second edition's numbering system *Fuller information on solutions to odd-numbered problems *Additional exercises and hints guide students in using the latest computer modeling tools Contents......Page 6 Preface to the Revised Second Edition......Page 10 Introduction......Page 16 1.1. Euclidean Space ......Page 18 1.2. Tangent Vectors ......Page 21 1.3. Directional Derivatives ......Page 26 1.4. Curves in R^3 ......Page 31 1.5. 1-Forms ......Page 38 1.6. Differential Forms ......Page 43 1.7. Mappings ......Page 49 1.8. Summary ......Page 56 2.1. Dot Product ......Page 58 2.2. Curves ......Page 67 2.3. The Frenet Formulas ......Page 73 2.4. Arbitrary-Speed Curves ......Page 84 2.5. Covariant Derivatives ......Page 96 2.6. Frame Fields ......Page 99 2.7. Connection Forms ......Page 103 2.8. The Structural Equations ......Page 109 2.9. Summary ......Page 114 3.1. Isometries of R^3 ......Page 115 3.2. The Tangent Map of an Isometry ......Page 122 3.3. Orientation ......Page 125 3.4. Euclidean Geometry ......Page 131 3.5. Congruence of Curves ......Page 136 3.6. Summary ......Page 143 4.1. Surfaces in R^3 ......Page 145 4.2. Patch Computations ......Page 154 4.3. Differentiable Functions and Tangent Vectors ......Page 164 4.4. Differential Forms on a Surface ......Page 173 4.5. Mappings of Surfaces ......Page 181 4.6. Integration of Forms ......Page 189 4.7. Topological Properties of Surfaces ......Page 199 4.8. Manifolds ......Page 208 4.9. Summary ......Page 216 5.1. The Shape Operator of M R^3 ......Page 217 5.2. Normal Curvature ......Page 224 5.3. Gaussian Curvature ......Page 231 5.4. Computational Techniques ......Page 239 5.5. The Implicit Case ......Page 250 5.6. Special Curves in a Surface ......Page 255 5.7. Surfaces of Revolution ......Page 267 5.8. Summary ......Page 277 6.1. The Fundamental Equations ......Page 278 6.2. Form Computations ......Page 284 6.3. Some Global Theorems ......Page 288 6.4. Isometries and Local Isometries ......Page 296 6.5. Intrinsic Geometry of Surfaces in R3 ......Page 304 6.6. Orthogonal Coordinates ......Page 309 6.7. Integration and Orientation ......Page 312 6.8. Total Curvature ......Page 319 6.9. Congruence of Surfaces ......Page 329 6.10. Summary ......Page 334 7.1. Geometric Surfaces ......Page 336 7.2. Gaussian Curvature ......Page 344 7.3. Covariant Derivative ......Page 352 7.4. Geodesics ......Page 361 7.5. Clairaut Parametrizations ......Page 368 7.6. The Gauss-Bonnet Theorem ......Page 379 7.7. Applications of Gauss-Bonnet ......Page 391 7.8. Summary ......Page 401 8.1. Length-Minimizing Properties of Geodesics ......Page 403 8.2. Complete Surfaces ......Page 415 8.3. Curvature and Conjugate Points ......Page 420 8.4. Covering Surfaces ......Page 431 8.5. Mappings That Preserve Inner Products ......Page 440 8.6. Surfaces of Constant Curvature ......Page 448 8.7. Theorems of Bonnet and Hadamard ......Page 457 8.8. Summary ......Page 464 Appendix: Computer Formulas ......Page 466 Bibliography ......Page 482 Answers to Odd-Numbered Exercises ......Page 483 Index ......Page 510 Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces.

The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard.

This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs Mathematica or Maple, as well as additional computer exercises. As with the Second Edition, this material supplements the content but no computer skill is necessary to take full advantage of this comprehensive text.

*Fortieth anniversary of publication! Over 36,000 copies sold worldwide
*Accessible, practical yet rigorous approach to a complex topic--also suitable for self-study
*Extensive update of appendices on Mathematica and Maple software packages
*Thorough streamlining of second edition's numbering system
*Fuller information on solutions to odd-numbered problems
*Additional exercises and hints guide students in using the latest computer modeling tools "Written primarily for students who have completed the standard first courses in calculus and linear algebra, Elementary Differential Geometry, Revised Second Edition, provides an introduction to the geometry of curves and surfaces ... This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs Mathematica and Maple, as well as additional computer exercises. This material supplements the content, and does not require previous experience with computers."--Back cover Intended for students who have completed the standard first courses in calculus and linear algebra, this edition provides an introduction to the geometry of curves and surfaces. It places emphasis on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard.