| Barrett O'Neill，加州大学洛杉矶分校教授。1951年在麻省理工学院获得博士学位。他的研究方向包括：曲线和曲面几何，计算机和曲面，黎曼几何，黑涧理论等。另著有Semi-Riemannian Geometry with Applications to Relativity和The Geometry of Kerr BlackHoles等书。... .. << 查看详细 |
| 1. calculus on euclidean space. 1.1. euclidean space 3 1.2. tangent vectors 6 1.3. directional derivatives 11 1.4. curves in r3 16 1.5. 1-forms 23 1.6. differential forms 28 1.7. mappings 34 1.8. summary 41 2. frame fields 2.1. dot product 43 2.2. curves 52 2.3. the frenet formulas 58 2.4. arbitrary-speed curves 69 2.5. covariant derivatives 81 2.6. frame fields 84 2.7. connection forms 88 2.8. the structural equations 94 2.9. summary 99 3. euclidean geometry .3.1. isometries of r3 100 3.2. the tangent map of an isometry 107 3.3. orientation 110 3.4. euclidean geometry 116 3.5. congruence of curves 121 3.6. summary 128 4. calculus on a surface 4.1. surfaces in r3 130 4.2. patch computations 139 4.3. differentiable functions and tangent vectors 149 4.4. differential forms on a surface 158 4.5. mappings of surfaces 166 4.6. integration of forms 174 4.7. topological properties of surfaces 184 4.8. manifolds 193 4.9. summary 201 5. shape operators.. 5.1. the shape operator of m c r3 202 5.2. normal curvature 209 5.3. gaussian curvature 216 5.4. computational techniques 224 5.5. the implicit case 235 5.6. special curves in a surface 240 5.7. surfaces of revolution 252 5.8. summary 262 6. geometry of surfaces in r3 6.1. the fundamental equations 263 6.2. form computations 269 6.3. some global theorems 273 6.4. isometries and local isometries 281 6.5. intrinsic geometry of surfaces in r3 289 6.6. orthogonal coordinates 294 6.7. integration and orientation 297 6.8. total curvature 304 6.9. congruence of surfaces 314 6.10. summary 319 7. riemannian geometry 7.1. geometric surfaces 321 7.2. gaussian curvature 329 7.3. covariant derivative 337 7.4. geodesics 346 7.5. clairaut parametrizations 353 7.6. the gauss-bonnet theorem 364 7.7. applications of gauss-bonnet 376 7.8. summary 386 8. global structure of surfaces 8.1. length-minimizing properties of geodesics 388 8.2. complete surfaces 400 8.3. curvature and conjugate points 405 8.4. covering surfaces 416 8.5. mappings that preserve inner products 425 8.6. surfaces of constant curvature 433 8.7. theorems of bonnet and hadamard 442 8.8. summary 449 appendix: computer formulas 451 bibliography 467 answers to odd-numbered exercises 468 index... 495 |
内容加载中，请稍后...
商品评论（0条）