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Oprea J. — Differential Geometry and Its Applications
Oprea J. — Differential Geometry and Its Applications

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Название: Differential Geometry and Its Applications

Автор: Oprea J.

Аннотация:

Designed not just for the math major but for all readers of science, this book provides an introduction to the basics of the calculus of variations and optimal control theory as well as differential geometry. It then applies these essential ideas to understand various phenomena, such as soap film formation and particle motion on surfaces.


Язык: en

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 387

Добавлена в каталог: 22.11.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Action integral      265
Agnesi      12
Alexandrov's Theorem      142
Analytic continuation      236
Arclength      4 43
Arclength, minimization      255 264 273
Area-minimizing      136—139 334
Area-minimizing, versus Enneper      248—249
Area-minimizing, versus Enneper, in Maple      251—253
Area-minimizing, via Pontryagin maximum principle      299
Astroid      10
Astroid, evolute of      35
Asymptotic curve      132
Bernstein's theorem      235
Bianchi identities      338
Binormal B      20
Bjorling's Problem      235—242
Bonnet's theorem      218
Brachistrochone      261—263 273
Catenary      13 139
Catenary, as solution to variational problem      283
Catenary, evolute of      34 46
Catenoid      59 139 182
Catenoid, as least area revolution surface      264
Catenoid, as least area revolution surface, via Pontryagin maximum principle      299
Catenoid, as minimal surface of revolution      114
Catenoid, as solution to Bjorling's problem      242
Catenoid, Gauss curvature of      101
Catenoid, mean curvature of      101
Catenoid, principal curvatures of      101
Catenoid, Weierstrass — Enneper Rep. of      228 231
Cauchy — Riemann equations      220
Cauchy — Schwarz inequality      see Schwarz's Inequality
Characterization of all umbilic surface      109
Characterization of Christoffel symbols by metric      327
Characterization of circle      25
Characterization of circular helices      37
Characterization of conformal Gauss maps      232
Characterization of constant H surfaces of revolution      115—119
Characterization of constant H surfaces of revolution, as roulettes of conies      119
Characterization of constant K surface      104 112
Characterization of curves by curvature and torsion      38
Characterization of curves by curvature and torsion, in Maple      see recreate3dview
Characterization of cylindrical helices      35
Characterization of extremals of surface area with fixed volume      293—294
Characterization of line      4 24
Characterization of line of curvature      81
Characterization of meridians and parallels as geodesies      161
Characterization of minimal surface of revolution      114
Characterization of minimal surface of revolution, as solution to Bjorling's problem      242
Characterization of nonarea minimizing minimal surfaces      245—246
Characterization of plane by shape operator      72
Characterization of plane curve      24
Characterization of plane curve in Maple      see recreate
Characterization of ruled minimal surfaces      132
Characterization of spherical curves      27 50
Characterization of tangent vector in $T_pM$      65
Christoffel symbols      106 327
Christoffel symbols, vanishing for $R^n$      327
circle      8 25
Circle, as solution to isoperimetric problem      283
Clairaut parameterization      160
Clairaut relation      159 163—164
Clairaut relation and D’ Alembert's principle      164
Codazzi — Mainardi equations      108
Complex differentiation      219—220
Complex integration      221
complex numbers      219
Cone      62 97—98
Cone, lassoing      179
Cone, unrolling as isometry      179
Conformal map      182
Conformal map, Gauss map on minimal surface      232
Conformal metric      168 291
Conjugate point      214—216
Connection 1—forms      352
Conservation of energy      265—266 289
Contraction      340—341
Contraction of metric      341
Contraction of Ricci      340
Coordinate patch (chart)      54 312
Covariant derivative      70—71 192 318
Covariant derivative of tensor R      338
Covariant derivative, properties      193 318
Cross product      19—20
Cross product in Maple      40
Curvature geodesic      151 172
Curvature geodesic, formula for      173 195
Curvature mechanical      292
Curvature normal      79—80 151
Curvature of a curve k      19
Curvature of a curve k and road banking      30
Curvature of a curve k in Maple      41
Curvature of a noriunit speed curve      28—30
Curvature of a noriunit speed curve, of involute      32
Curvature principal      81
Curvature Ricci      340
Curvature Riemann      335—336
Curvature scalar      340
Curvature sectional      337
Curvature sectional of sphere      337
Curvature, Gauss K      87 105 168
Curvature, Gauss K, constant      102—104 112
Curvature, Gauss K, in Maple      121
Curvature, line of      81 93 94
Curvature, mean H      87 325
Curvature, mean H in Maple      121
Curve      1
Curve asymptotic      132
Curve Newton's      3
Curve of constant precession      23 51—52
Curve parameter      53
Curve speed of      3
Curve spherical      27 50—51
Curve spherical, total torsion of      80
Curve, parameterized line      1
Curve, velocity vector of      1—2
Cusp      16
Cycloid      9—10
Cycloid, as brachistrochone      263
Cycloid, as tautochrone      10
Cylinder      62 82 97—98
Cylinder, shape operator of      72
Cylinder, unrolling      180
Cylinder, versus Enneper in area      248—249
Cylinder, versus Enneper in area, in Maple      251—253
Darboux vector      22—23 37
delaunay      see surface
Derivative mapping      76—77
Developable surface      98
Developable surface and lines of curvature      98
Developable surface, tangent developable      98
Direction vector of line      2
Directional derivative      69
Divergence      141 346
Divergence of metric      347
Divergence, Theorem      141
Dot product      5 167
Dot product in Maple      40
D’Alembert's principle      164 284
Eigenvalue      74
Eigenvalue and determinant      74
Eigenvalue and trace      74
Eigenvector      74
Einstein curvature G      345 349—350
Einstein manifold      349
Elastic rod      282
ellipse      15 17 31
Ellipse, evolute of      34
Ellipse, in Maple      45
Enneper's surface      61 85 229
Enneper's surface, curvatures of      93
Enneper's surface, non area minimizing      248—249
Enneper's surface, non area minimizing, in Maple      251—253
Euler characteristic      205
Euler characteristic, formula for plane      206
Euler — Lagrange equations      257 280
Euler — Lagrange equations two-function      257—258
Euler — Lagrange equations two-variable      258
Euler — Lagrange equations, in Maple      300—304
Euler — Lagrange equations, independence from t      259
Euler — Lagrange equations, with undetermined endpoint      260
Euler, formula of      84
Euler, formula of and mean curvature      90
Euler, spiral of      38
Euler, spiral of and minimizing bending energy      281
Evolute      32 37 45
Evolute of astroid      35
Evolute of astroid in Maple      46
Evolute of catenary      34 46
Evolute of ellipse      34
Evolute of ellipse in Maple      45
Evolute of parabola      33
Extremal      257 267
Fixed endpoint problem      255
Flat      90 115 336
Flat, surface      90
Flat, torus      171
Forms      351
Foucault's pendulum      198—200
Foucault's pendulum, as parallel vector field      199
Frame $\epsilon_i$      191 195 339 352
Frame $\epsilon_i$ dual      352 356
Frenet formulas      21
Frenet formulas for nonunit speed curves      28
Frenet frame      20
Gauss map      77—78
Gauss map and area minimization      246
Gauss map and Gauss curvature      90
Gauss map and shape operator      77—78
Gauss map, as meromorphic function      233
Gauss map, conformality of      183 232
Gauss — Bonnet Theorem global      205
Gauss — Bonnet Theorem local      203 214
Gauss(ian) curvature      87 336
Gauss(ian) curvature and compactness      112
Gauss(ian) curvature and Gauss map      90
Gauss(ian) curvature and Jacobi equation      212
Gauss(ian) curvature and second derivative test      98—99
Gauss(ian) curvature and shape operator      88—89
Gauss(ian) curvature from Weierstrass — Enneper Rep.      231
Gauss(ian) curvature in Maple      121
Gauss(ian) curvature of a surface of revolution      100
Gauss(ian) curvature of Enneper's surface      93
Gauss(ian) curvature of hyperboloid of two sheets      95—96
Gauss(ian) curvature of ruled surface      97
Gauss(ian) curvature of sphere      91 95 108 357
Gauss(ian) curvature of torus      100—101
Gauss(ian) curvature, formula for      89 91 105 168
Gauss(ian) curvature, from dual 1-form frame      356
Gauss(ian) curvature, metric formula for      105
Gauss(ian) curvature, Theorem Egregium      105
Gauss(ian) curvature, total, and holonomy      190—191 197
Geodesic      151 192
Geodesic and Maple      185—186
Geodesic and Pontryagin's maximum principle      297—298
Geodesic closed, as periodic orbit      292
Geodesic curvature      151 172
Geodesic curvature, formula for      173 195
Geodesic equations      156 185 329
Geodesic equations, as Euler — Lagrange equations      288 330
Geodesic equations, for Clairaut patch      160
Geodesic of Poincare and hyperbolic planes      181
Geodesic on cone      163
Geodesic on cylinder      154 156
Geodesic on hyperbolic plane      175
Geodesic on hyperboloid of one sheet      165
Geodesic on paraboloid      165
Geodesic on Poincare plane      174 187
Geodesic on sphere      153—154 157
Geodesic on stereographic sphere      176
Geodesic on torus      159 186
Geodesic on whirling witch of Agnesi      163 186
Geodesic torsion      195
Geodesic, as equation of motion      286
Geodesic, as equation of motion in Maple      306—307
Geodesic, as equation of motion with conformal metric      291
Geodesic, as line of curvature      154
Geodesic, as shortest distance      155 288
Geodesic, as shortest distance, detected by Jacobi equation      217 275
Geodesic, existence and uniqueness of      157
Geodesic, invariant under isometry      180
Geodesic, polar coordinates      155 209
Geodesically complete      166 217
Green's theorem      129 138 140 201
Hamilton's principle      265 286
Hamiltonian      296
Harmonic function      146
Harmonic function, as Euler — Lagrange equation      265
Helicoid      60 97 178 182
Helicoid, Weierstrass — Enneper Rep. of      228
Helix      12 18 24 42
Helix circular      37
Helix cylindrical      35
Helix hyperbolic      32
Hilbert's invariant integral      269
Hilbert's invariant integral, independence of path of      269—271
Hilbert's Lemma      114
Holomorphic      219
Holonomic constraints      284
Holonomy      195
Holonomy along latitude of sphere      196
Holonomy along parallel of cone      198
Holonomy and Foucault's pendulum      198—200
Holonomy on Poincare plane      201
Holonomy, as total Gauss curvature      197
Hyperboloid of one sheet      63
Hyperboloid of one sheet, closed geodesies on      165 207
Hyperboloid of one sheet, Gauss curvature of      97
Hyperboloid of two sheets      95
Hyperboloid of two sheets and Bonnet's theorem      218
Hyperboloid of two sheets Gauss curvature of      96
Hypersurface      317 342
Identity Theorem      236
Inner product      167—168
Involute      17—18 44
Involute, curvature of      32
Isometry      177
Isometry global      178
Isometry local      177
Isometry unrolling a cone      179
Isoperimetric problem      283—284
Isoperimetric problem, Euler — Lagrange equation for      283
Isothermal coordinates      148
Isothermal coordinates, existence of on minimal surfaces      222
Jacobi equation      212 216 275
Jacobi equation on sphere      216
Jacobi equation, as Euler — Lagrange equation      275—277
Jacobi's theorem      291
Jacobian      77 316
Jacobian of sphere      316—317
kinetic energy      255 265 284
Kuen's surface      97
Lagrange's identity      19 89
Laplace — Young equation      127
Lie bracket      319
Lie group      344
Liebmann's Theorem      112
Line      1—3 150
Line of curvature      81 93 94
1 2
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