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Kühnel W., Hunt B. — Differential Geometry: Curves - Surfaces - Manifolds

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Название: Differential Geometry: Curves - Surfaces - Manifolds

Авторы: Kühnel W., Hunt B.

Аннотация:

Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in \$I\!\!R^3\$ that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to helpalong the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Издание: Second Edition

Год издания: 2006

Количество страниц: 376

Добавлена в каталог: 30.04.2008

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Предметный указатель
 Acceleration vector      141 angle      2 Angle preserving      101 128 Apex      12 Arc element      60 Arc length      9 Archimedean spiral      51 Asymptotic curve      84 128 Atlas      204 Banchoff, T.      187 Beltrami, E.      85 95 Bertrand curve      53 Bianchi identity      249 341 Binary dihedral group      309 Binary icosahedral group      309 Binary octahedral group      309 Binary tetrahedral group      309 Binormal      17 Biquadratic form      252 Bivector      337 Bonnet, O.      154 Boost      276 Canal surface      78 Cardan angles      206 Cartan, E.      166 Catalan, E. C.      113 129 Catenary      11 112 195 Catenoid      112 157 196 Cauchy-Riemann equations      103 Cayley map      205 Cayley plane      336 Chain rule      214 Chart      5 202 203 Christoffel symbols      139 160 168 228 circle      9 Clifford torus      32 Codazzi-Mainardi equation      147 154 170 198 248 Cohn-Vossen, S.      190 Complex manifold      207 Complex projective space      335 Complex structure      209 Cone      90 91 93 conformal      101 105 107 128 221 278 Conformal curvature      346 Conformally flat      126 351 353 Conic type      82 Conjugate point      312 Conjugate surface      112 Connection      226 Connection form      168 Constant curvature      25 36 81 191 193 255 273 294 316 Constant Gaussian curvature      162 Contact of kth order      12 Contraction      257 Contravariant tensor      242 Convergence      208 Convex      42 183 Convex hull      45 183 Coordinate transformation      204 Cornu spiral      16 Cosmological constant      331 Countability axiom      222 Covariant derivative      136 138 168 226 245 272 Covariant tensor      242 Covector field      243 Covering      298 307 316 Coxeter, H.S.M.      308 CR equations      103 Cubical parabola      20 Curvature      14 17 20 36 71 Curvature tensor      150 171 249 253 321 Curve      7 8 Curve, closed      37 Curve, length of      8 Curve, simply closed      37 45 Cyclic group      307 309 Cycloid      50 Cylinder      90 91 Darboux equations      26 52 Darboux vector      26 52 Derivative      3 6 214 Developable surface      89 121 Dicyclic group      309 Diffeomorphic      208 Differentiable      3 Differentiable manifold      203 Differentiable structure      204 Differential      6 214 246 Differential form      167 Dihedral group      307 Dini, U.      95 Directional derivative      100 135 210 211 226 Directional vector      63 Directrix      85 Distance      2 Divergence      257 258 Double point      48 Double tangent      48 Dual basis      217 Duality      356 Dupin indicatrix      76 Eigenvalue      72 Eigenvector      72 Einstein field equations      330 Einstein space      261 334 357 Einstein tensor      263 330 Einstein, A.      318 323 357 ellipse      76 Ellipsoid      131 Elliptic point      73 196 Elongated sphere      82 Energy functional      284 Enneper, A.      85 113 Equations of Gauss and weingarten      140 146 Euler angles      206 Euler characteristic      179—181 190 222 Evolute      15 Exponential mapping      231 285 Exterior derivative      168 Fabricius-Bjerre, Fr.      48 Fenchel, W.      46 Fermi coordinates      161 First fundamental form      59 Flow      235 Focal curve      15 Four Vertex Theorem      45 Free motion      275 Frenet curvature      27 Frenet curve      13 Frenet equations      14 17 36 Frenet matrix      26 27 Frenet n-frame      13 Frobenius, G.      155 Gauss equation      147 150 154 170 198 267 272 365 Gauss formula      140 Gauss lemma      287 Gauss map      64 67 Gauss, C. F.      148 326 Gauss-Bonnet formula      173 326 Gauss-Kronecker curvature      125 Gaussian curvature      73 119 148 195 248 Geodesic      72 123 141 220 229 230 285 312 Geodesic coordinates      277 Geodesic curvature      23 72 127 172 Geodesic parallel coordinates      160 Geodesic polar coordinates      288 295 Geodesic torsion      127 Geodesic triangle      177 Geometric linearization      8 55 Golden ratio      308 Gradient      246 319 Gram determinant      183 Gram-Schmidt orthogonalization      13 Graph      58 74 Groups of motion      274 Harmonic function      102 Hausdorff separation axiom      208 Helicoid      87 112 157 Helicoidal ruled surface      87 95 Helix      9 20 Henneberg, L.      113 Hesse tensor      246 hessian      74 246 350 Hessian matrix      74 Hexagonal torus      299 Hilbert, D.      192 318 323 Hilbert-Einstein functional      318 Hodge operator      356 359 Holomorphic      103 107 Holonomy group      232 Homogeneous space      332 Hopf, H.      41 181 Hyperbola      35 76 Hyperbolic plane      121 122 198 Hyperbolic point      73 Hyperbolic space      271 273 297 Hyperboloid      74 84 86 115 129 271 Hyperboloid type      82 Hyperplane      126 Hypersphere      126 Hypersurface element      124 Icosahedral group      307 icosahedron      308 Immersion      3 5 8 55 Implicit function      3 INDEX      126 Index form      282 Inflection point      14 48 Inner product      2 218 Instantaneous speed      8 Integrability conditions      146 147 150 155 170 353 Inverse mapping      4 Irreducible      333 Isometric      162 Isometry      221 295 Isometry group      332 Isothermal      101 Isotropic      34 116 132 Isotropy group      332 Jacobi determinant      65 Jacobi equation      289 Jacobi field      289 291 312 Jacobi identity      224 249 Jacobian      3 Klein bottle      205 222 Koszul formula      349 Kuiper, N. H.      187 Lagrange multiplier      72 Laplace-Beltrami operator      258 Laplacian      258 Length preserving      162 Lens space      310 314 Level point      73 106 107 Levi-Civita connection      226 Lie algebra      232 Lie bracket      138 223 236 Lie derivative      224 Lie group      232 Lie, S.      223 Liebmann, H.      46 191 193 Light-cone      34 115 Light-like      34 Light-like line      35 Line      9 85 123 Line of curvature      77 Lines of curvature parameters      77 105 Locally compact      208 Locally isometric      256 Logarithmic spiral      51 Lorentz group      274 Lorentz rotation      119 Lorentz space      34 274 Lorentz transformation      276 Lorentzian metric      219 222 Manifold      203 Maurer-Cartan equations      170 Mean curvature      73 99 125 130 Mean curvature vector      102 Measure tensor      240 Mercator projection      128 Meridian curve      78 Meromorphic      107 209 Metric tensor      240 Meusnier, M.      72 Minimal surface      99 132 Minkowski space      33 114 219 274 Mobius strip      65 Mobius, A.      66 Monge coordinates      74 Monge surface      132 Monge, G.      132 Monkey saddle      74 Multilinear      242 Multiplicity      312 Neil parabola      20 Non-Euclidean geometry      122 Norm      2 Normal coordinates      285 Normal curvature      72 172 Normal plane      20 Normal section      72 Normal space      6 56 Normal variation      98 Normal vector      14 57 Null vector      34 270 Null-cone      115 Oblate sphere      82 Octahedral group      307 Octahedral space      310 octahedron      308 Orientability      64 Orthogonal group      205 274 Osculating plane      20 72 Osculating sphere      21 Ovaloid      184 parabola      20 Parabola of contact      49 Parabolic point      73 Paraboloid      74 129 parallel      141 229 230 Parallel displacement      142 230 Parallel surface      130 PARAMETER      56 Parameter of distribution      87 Parameter transformation      65 Parametrization      5 56 202 Parametrized curve      8 Partition of unity      221 Petrov type      364 Pfaffian form      167 Poincare upper half-plane      196 220 312 Polar angle      42 Polar angle function      38 Polar coordinates      37 277 Polarization      252 Position vector      63 Potential equation      351 primitive      105 Principal curvature      72 125 259 265 Principal normal      17 195
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