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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:

Differential geometry has a long, wonderful history it has found relevance in areas ranging from machinery design of the classification of four-manifolds to the creation of theories of natures fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole, it mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors, it is also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only see geodesics on surfaces, but they will also see the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. Students will also see how particles move under constraints. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 2007

Количество страниц: 469

Добавлена в каталог: 11.12.2011

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$k$-patch      397
1-forms      430
2-body problem      360
2-cell embedding      291
2-forms      432
2-independent variable EL equation      316 326
2-variable EL equation      315
2-variable first integral      316
Acceleration components      210
Acceleration formulas      125
Acceleration vector      3
Action integral      326 358
Adjoint      193
Adjoint of minimal surface      193
Adjoint to Henneberg's surface      193
Alexandrov's Theorem      176
Almost complex structure J      230
Angle excess theorem      289 301
angular momentum      223
Arclength      4 214 222
Arclength of geodesic on cone      248
Arclength parametrization      14
Area      414
Area minimization and Maple      203
Area minimization implies minimal      172 416
Area of bugle surface      277
Area of pseudosphere      277
Area of sphere      276
Area of surface      164 276 277
Area of torus      277
Area variation      171
Associated family of minimal surfaces      193
Astroid      10 11 33
Astroid evolute      33
Asymptotic curve      166
Banked highway      29
BAT      207
Bending energy      347
bernoulli      324
Bernoulli's principle      330
Bernstein's theorem      197
Bianchi identity      419
Binormal      19
Bonnet's theorem      304
Bonnet's theorem, converse      305
Book Cover      207
Brachistochrone problem      322
Brachistoclirone      341
Bracket      403
Buckled column      330
Bump lemma      314
Cantilevered beam      332
Catalan's surface      168
Catalan's surface, via Maple      198
Catalan's surface, WE representation      190
Catalan's theorem      167
Catenary      12 13 173 349
Catenary evolute      33
Catenoid      73 120 132 325 370
Catenoid, associated family      193
Catenoid, via Maple      197 201
Catenoid, WE representation      190
Catenoid-helicoid deformation      240
Cauchy - Riemann equations      183
Christoffel symbols      125 409
Christoffel symbols, determined by metric      410
circle      8 14 16 20 24 350
Circle, characterization of      24
Circle, osculating      26
Circular helix      35 36
Clairaut geodesic equations      220
Clairaut parametrization      219
Clairaut relation      218 222 224 225 353
Clairaut relation, on torus      219
Clairaut relation, physical viewpoint      223
Closed      129
Co-state equations      368 369
Codazzi - Mainardi equations      127
Column, buckled      330
Compact      128
complete      225
Complex analytic      182
Complex conjugate      183
Complex differentiable      182
Complex integral      184
Cone      75 117
Cone unrolling      236
Cone unrolling, via Maple      249
Conformal Gauss map      193-195
Conformal map      194 240 241
Conformal metric      227 360
Conformal metric, scaling factor      227
Conformality factor      194
Conjugate point      302 303 344
Connected      69
Connection 1-forms      431
Constant Gauss curvature      112 121 123
Constant Gauss curvature, via Maple      153
Constant mean curvature      112 134 136 137 366
Constant precession curve      21
Constant speed relation      216
Constrained problem      346 347 349 365 382
Contraction      422
Contraction of metric      422
Coordinate chart      397
Covariant derivative      82 278 286 402 423 431
Covariant derivative, properties of      279 402
Cross product      18
CrossProduct      43
Curvature of curve      17 23 29 43 329 347
Curvature of involute      31
Curvature of plane curve      17
Curvature, 2-form      434
Curvature, average normal      109
Curvature, constant Gauss      112 121 123
Curvature, constant mean      112 134 137
Curvature, Einstein      425
Curvature, Gauss      107 109
Curvature, geodesic      210
Curvature, line of      93
Curvature, mean      107 109 181
Curvature, nonnegative Gauss      305
Curvature, normal      92
Curvature, principal      93 109
Curvature, Ricci      420
Curvature, Riemann      417
Curvature, scalar      421
Curvature, sectional      418
Curvature, total Gauss      110 277
Curve      1
Curve of constant precession      21 63
Curve, arclength of      4
Curve, characterization of      38
Curve, closed      38 69
Curve, cusp      3
Curve, differentiable      1
Curve, evolute of      31
Curve, non-unit speed      27
Curve, rectifying      36
Curve, regular      3
Curve, simple      39
Curve, smooth      1
Curve, speed of      3
Curve, torsion of      20
Curve, total torsion of      22
Cycloid      9 51 324
Cylinder      75 94 117
Cylinder unrolling      237 249
Cylinder, twisted      105
Cylindrical helix      34 36
Cylindrical helix, characterization of      34
D'Alembert's principle      223 350
Darboux vector      21 36
Delaunay surface      133 137 365
Delaunay theorem      137
Derivative map      89
Developable surface      117 263
Direction vector      2
Directional derivative      81
Directrix      75
Dirichlet integral      326
Divergence      426
Divergence of metric      426
Divergence of Ricci curvature      427
Dot product      5
DotProduct      43
Double pendulum      363
Doubly ruled      75
Douglas - Rado theorem      171
Eigenvalue      86
Einstein curvature      425
Einstein manifold      428
Elastic rod      348
ellipse      14 16 30
Ellipse evolute      33
Ellipse, arclength of      140
Elliptic E correction      xv 145 393
Elliptic functions      138
Elliptic integral      138
Endpoint-curve problem      318
Enneper's surface      74 97 112 170
Enneper's surface, via Maple      198
Enneper's surface, WE representation      191
Euler - Lagrange equation      314-316 319 328 347 355 358 367 380 412
Euler characteristic      291 292
Euler's formula      96
Euler's spiral      38 348
Evolute      31 48
Evolute of astroid      33 49
Evolute of catenary      33 49
Evolute of ellipse      33 48
Evolute of parabola      32
Exterior derivative      432
Extremal      315 334 356
Extremize      315 356
Fermat's principle      324
Feynman quote      284
Field of extremals      334 345
Final time fixed      317
First Bianchi identity      419
First integral      316 358 381
First structure equation      432
Fixed endpoint problem      312 316 341
Flat      417
Flat surface      110
Flat surface of revolution      133
Flat surfaces of revolution via Maple      201
Flat torus      229 230
Foucault pendulum      284
Foucault vector field      285
Frame      278
Frame field      278 420 434
Frenet formulas      20 28
Frenet frame      19
fundamental frequency      365
Fundamental theorem of space curves      38
Gauss - Bonnet theorem      291
Gauss curvature      107 109 227 418 434
Gauss curvature of $R$-sphere      114 126
Gauss curvature of Enneper's surface      114
Gauss curvature of minimal surface      193
Gauss curvature of surface of revolution      119 121
Gauss curvature, depends only on metric      124
Gauss curvature, nonnegative      305
Gauss curvature, sign of      108
Gauss curvature, via Maple      150
Gauss map      90 110 193 194
Gauss map, area of      110
Gauss map, for catenoid      91
Gauss map, for cone      90
Gauss map, for cylinder      90
Gauss map, for Enneper      91
Gauss map, from WE representation      195
Gauss's lemma      298
Geodesic      212 238 352 356 369 411
Geodesic constant speed relation      216
Geodesic curvature      210 231 281 358
Geodesic curvature, depends only on metric      211
Geodesic equations      216 369 413
Geodesic equations, Clairaut      220
Geodesic equations, via Maple      242
Geodesic has constant speed      212
Geodesic in conformal metric      256
Geodesic on cone      222
Geodesic on cone, via Maple      247
Geodesic on cylinder      214 216
Geodesic on hyperbolic plane      233 239
Geodesic on hyperboloid of 1-sheet      225 293
Geodesic on paraboloid      224
Geodesic on plane      222
Geodesic on Poincare plane      232
Geodesic on sphere      213 216
Geodesic on sphere, via Frenet formulas      214
Geodesic on stereographic plane      234 256
Geodesic on stereographic sphere      234
Geodesic on surface of revolution      221
Geodesic on torus      218
Geodesic on unduloid      251
Geodesic on whirling witch of Agnesi      222
Geodesic parameter curve      221
Geodesic plane curves      216
Geodesic polar coordinates      214 297
Geodesic torsion      281 282
Geodesic via Maple      243
Geodesic, as length minimizer      214
Geodesic, as line of curvature      214
Geodesic, existence of      216
Geodesically complete      225
Geodesics      353
Geographical coordinates      70
Goldschmidt discontinuous solution      173
Great circle      213
Green's theorem      39 287
Hadamard's Theorem      296
Hamilton's principle      312 326
Hamiltonian      368 369
Harmonic conjugate      183
Harmonic function      179
Helicoid      74 75 116 166
Helicoid isometry      236
Helicoid, WE representation      190
Helicoid-catenoid animation      199
Helix      11 16 44
Helix, circular      35 56
Helix, curvature of      22
Helix, cylindrical      34
Helix, hyperbolic      31
Helix, involute of      17
Henneberg's surface      168
Henneberg's surface adjoint      193
Henneberg's surface, WE representation      191
Higher-order Euler - Lagrange equation      329
Hilbert's invariant integral      337
Hilbert's Lemma      132
Holomorphic      182
Holonomic constraints      350
Holonomy      281 283
Holonomy, as total Gauss curvature      288
Holonomy, on cone      284 308
Holonomy, on Poincar$\acute{e}$ plane      287
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