| Preface |
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1 | (56) |
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1 | (5) |
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1.2 Linear transformations |
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6 | (6) |
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12 | (7) |
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19 | (6) |
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1.5 Subspaces of Euclidean space |
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25 | (2) |
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1.6 Determinants as volume |
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27 | (3) |
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1.7 Elementary topology of Euclidean spaces |
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30 | (6) |
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36 | (5) |
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1.9 Limits and continuity |
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41 | (7) |
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48 | (9) |
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57 | (60) |
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57 | (5) |
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2.2 Basic properties of the derivative |
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62 | (5) |
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2.3 Differentiation of integrals |
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67 | (2) |
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69 | (6) |
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2.5 The inverse and implicit function theorems |
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75 | (6) |
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2.6 The spectral theorem and scalar products |
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81 | (8) |
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2.7 Taylor polynomials and extreme values |
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89 | (5) |
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94 | (9) |
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103 | (5) |
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108 | (2) |
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110 | (7) |
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117 | (60) |
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3.1 Submanifolds of Euclidean space |
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117 | (7) |
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3.2 Differentiable maps on manifolds |
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124 | (5) |
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3.3 Vector fields on manifolds |
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129 | (8) |
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137 | (4) |
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141 | (2) |
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3.6 Covariant differentiation |
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143 | (5) |
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148 | (5) |
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3.8 The second fundamental tensor |
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153 | (3) |
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156 | (4) |
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160 | (3) |
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163 | (5) |
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168 | (9) |
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4 Integration on Euclidean space |
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177 | (44) |
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4.1 The integral of a function over a box |
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177 | (4) |
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4.2 Integrability and discontinuities |
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181 | (6) |
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187 | (8) |
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195 | (3) |
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4.5 The change of variables theorem |
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198 | (4) |
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4.6 Cylindrical and spherical coordinates |
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202 | (8) |
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4.6.1 Cylindrical coordinates |
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202 | (4) |
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4.6.2 Spherical coordinates |
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206 | (4) |
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210 | (4) |
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211 | (1) |
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211 | (2) |
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213 | (1) |
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214 | (7) |
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221 | (46) |
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5.1 Tensors and tensor fields |
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221 | (3) |
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5.2 Alternating tensors and forms |
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224 | (8) |
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232 | (4) |
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5.4 Integration on manifolds |
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236 | (4) |
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5.5 Manifolds with boundary |
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240 | (3) |
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243 | (3) |
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5.7 Classical versions of Stokes' theorem |
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246 | (6) |
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5.7.1 An application: the polar planimeter |
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249 | (3) |
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5.8 Closed forms and exact forms |
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252 | (5) |
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257 | (10) |
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6 Manifolds as metric spaces |
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267 | (34) |
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6.1 Extremal properties of geodesies |
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267 | (4) |
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271 | (4) |
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6.3 The length function of a variation |
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275 | (3) |
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6.4 The index form of a geodesic |
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278 | (5) |
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6.5 The distance function |
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283 | (2) |
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6.6 The Hopf-Rinow theorem |
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285 | (4) |
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289 | (3) |
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292 | (9) |
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301 | (38) |
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7.1 Hypersurfaces and orientation |
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301 | (3) |
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304 | (4) |
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7.3 Curvature of hypersurfaces |
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308 | (5) |
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7.4 The fundamental theorem for hypersurfaces |
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313 | (3) |
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7.5 Curvature in local coordinates |
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316 | (2) |
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7.6 Convexity and curvature |
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318 | (2) |
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320 | (3) |
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7.8 Surfaces of revolution |
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323 | (5) |
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328 | (11) |
| Appendix A |
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339 | (6) |
| Appendix B |
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345 | (6) |
| Index |
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351 | |
Lisainfo e-raamatute kohta