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O'Neill B. — Elementary differential geometry
O'Neill B. — Elementary differential geometry



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Íàçâàíèå: Elementary differential geometry

Àâòîð: O'Neill B.

ßçûê: en

Ðóáðèêà: Ìàòåìàòèêà/Ãåîìåòðèÿ è òîïîëîãèÿ/Äèôôåðåíöèàëüíàÿ ãåîìåòðèÿ/

Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö

ed2k: ed2k stats

Ãîä èçäàíèÿ: 1966

Êîëè÷åñòâî ñòðàíèö: 411

Äîáàâëåíà â êàòàëîã: 20.04.2005

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
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Ïðåäìåòíûé óêàçàòåëü
$\ell$, m, n      211—213
1-segment      167
2-segment      169
Acceleration      54 68
Acceleration of a curve in a surface      196 324(Ex.
Acceleration, intrinsic      320
Adapted frame field      246—251
Algebraic area      289—290 386(Ex.
All-umbilic surface      257—259
Alternation rule      27 47 153
angle      44
Angle function      50(Ex. 12) 295(Ex.
Angle, coordinate      210
Angle, exterior      374
Angle, interior      374
Angle, oriented      291
Antipodal mapping      165(Ex. 5)
Antipodal points      182
Arc length      51—52 220(Ex.
Arc length function      51
Area      280—286
Area form      283—284 292 309(Ex.
Area-preserving mapping      295(Ex. 16)
Associated frame field      276—277
Asymptotic curves      226—227 230
Asymptotic directions      225—227
Attitude matrix      46 88
Basis formulas      251
Bending      268 275(Ex.
binomial      57 66 69
Boundary of a 2-segment      170
Boundary of a region      386
Bracket operation      81(Ex. 7) 195(Ex.
Bugle surface      241—242 244(Ex. 282—283 291
Canonical isomorphism      7—8 44 59
Canonical parametrization      238
Cartan, E.      42 92 96 303
Cartesian Plane      306
Cartesian product      187(Ex. 9)
Catenoid      236
Catenoid as minimal surface      236—238
Catenoid, Gauss mapping      296(Ex. 20)
Catenoid, Gaussian curvature      236 239
Catenoid, Gaussian curvature, total      287—288
Catenoid, local isometry onto      267—268
Catenoid, principal curvatures      236
Center of curvature      64(Ex. 6)
circle      59 62
Clairut parametrization      330—339
Closed surface in $E^{3}$      181(Ex. 10) 263n
Codazzi equations      249 255
Compact surface      176—177 259—260
Complete surface      see "Geodesically complete surface"
Composite function      1
Cone      140(Ex. 5) 231(Ex.
Conformal geometric structure      305—306 308(Ex. 312—313
Conformal mapping      268—271 310
Conformal patch      270(Ex. 7) 279(Ex.
Congruence of curves      116—123
Congruence of curves determined by curvature and torsion      117
Congruence of surfaces      189 297—303
Conjugate point      353—359
Connected surface      176
Connection equations on a surface      248 318
Connection equations on Euclidean space      86 248
Connection forms on a surface      248 272 277 306
Connection forms on Euclidean space      85—90
Conoid      233(Ex. 20)
Consistent formula for a mapping      165(Ex. 10)
Coordinate angle      210 220(Ex.
Coordinate expression      143 165(Ex.
Coordinate patch      see "Patch"
Coordinate system      158(Ex. 9) 276—277
Covariant derivative formula      91(Ex. 5) 189—190 321
Covariant derivative on a patch      212 324—326
Covariant derivative, Euclidean      77—80 116(Ex. 189—190
Covariant derivative, intrinsic      318—326
Cross product      47—49 107 110
Cross-sectional curve      138
Curvature      see also "Gaussian curvature"
Curvature of a curve in $E^{2}$      65(Ex. 8) 122—123
Curvature of a curve in $E^{3}$      57 66 69
Curve      15
Curve in a surface      144—145
Curve plane      61
Curve segment      56(Ex. 10) 167
Curve, closed      169
Curve, coordinate expression for      144
Curve, one-one      20
Curve, periodic      20
Curve, regular      20
Curve, simple closed      151n
Curve, unparametrized      20—21
Cylinder      129 231(Ex.
Cylinder parametrization      137 141(Ex.
Cylinder, geodesies      275(Ex. 2) 352(Ex.
Cylindrical frame field      82—83
Cylindrical frame field, connection forms      89—90
Cylindrical frame field, dual 1-forms      96(Ex. 4)
Cylindrical helix      72—75
Darboux      81
Degree of a form      27 152
Degree of a mapping      386(Ex. 6)
Derivative map      35—40
Derivative map of a mapping of surfaces      160—161 166
Derivative map of a patch      149(Ex. 4)
Derivative map of an isometry      104
Derivative of Euclidean vector field      113
Diffeomorphism      38 40(Ex. 161
Differentiability      4 10 33 143 145—146
Differential      23—26
Differential form on $E^{2}$      156
Differential form on $E^{3}$      22—31
Differential form on a surface      152—157
Differential form, closed or exact      157(Ex. 2) 173—175
Differential form, pullback      163
Direction      196
Directional derivative      11—15 149
Directional derivative, computation of      12 25
Director curve      140
Distribution parameter      232(Ex. 14)
Domain      1
Dot product      42—44 46 82 210
Dot product, preserved by isometries      105
Dual 1-forms      91—92 248 306
Dupin curves      208(Ex. 5)
E, F, G      140(Ex. 2) 210 317(Ex.
Edge (curve)      170 373 377
Ellipsoid      142(Ex. 10)
Ellipsoid isometry group      365
Ellipsoid, Euclidean symmetries      303(Ex. 10)
Ellipsoid, Gaussian curvature      217—219 222(Ex.
Ellipsoid, umbilics      223(Ex. 23)
Elliptic hyperboloid      142(Ex. 10) 221—222
Elliptic paraboloid      142(Ex. 11) 220(Ex.
Enneper's surface      221(Ex. 12)
Euclid      335—336
Euclidean coordinate functions      9 15 24 33 53
Euclidean distance      43 49(Ex.
Euclidean geometry      112 304—305 308
Euclidean plane      5 305 335—336
Euclidean plane, local isometries      363—364
Euclidean space      3 5
Euclidean symmetry      302—303 365
Euclidean vector field      147
Euler — Poincare characteristic      378—380
Euler's formula      201
Evolute      75(Ex. 15)
Exterior angle      374
Exterior derivative      28—31 31—32 154—155
Faces      377
Flat surface      207 231(Ex. 263(Ex.
Flat torus      316 317(Ex. 370(Ex.
Flat torus imbedding in $E^{4}$      369
Focal point      362
FORM      see "Differential form"
Frame      44
Frame field on $E^{3}$      82
Frame field on a curve      120
Frame field on surface      251 306
Frame field, adapted      246
Frame field, natural      9
Frame field, principal      254
Frame field, transferred      272—273
Frame-homogeneous surface      366
frenet      81
Frenet apparatus      63(Ex. 1) 71
Frenet approximation      60—61 65(Ex.
Frenet formulas      58 67
Frenet frame field      57
Function      1—2
Function, one-to-one      2
Function, onto      2
Gauss      245 274 303
Gauss equation      249
Gauss mapping      194—195 289—291
Gauss — Bonnet formula      375—377 383—385 388(Ex.
Gauss — Bonnet theorem      380—383 387(Ex.
Gaussian curvature      203—207 310—312 see
Gaussian curvature in Jacobi equation      355
Gaussian curvature sign      204—205
Gaussian curvature, differentiability      206
Gaussian curvature, formulas, explicit      206 212 217 253 278
Gaussian curvature, formulas, implicit      205 252
Gaussian curvature, Gauss mapping and      289
Gaussian curvature, geodesic curvature an      339(Ex. 19)
Gaussian curvature, geodesies and      355—358
Gaussian curvature, holonomy and      325(Ex. 5)
Gaussian curvature, interval      275(Ex. 3)
Gaussian curvature, isometric invariance      273—275
Gaussian curvature, polar circles and      359—360
Gaussian curvature, polar discs and      361(Ex. 2)
Gaussian curvature, principal curvatures and      203
Gaussian curvature, shape operator and      203
Geodesic curvature      230(Ex. 7) 329—330 337(Ex.
Geodesic curvature, total      372—373
Geodesic polar mapping      340—341
Geodesic polar parametrization      341—344
Geodesically complete surface      263n 328—329
Geodesically complete surface, shortest geodesic segments in      348
Geodesies      228—232 326—363 see
Geodesies coordinate formulas      327 333
Geodesies on surfaces of nonpositive curvature      358 388(EX.
Geodesies, closed      232(Ex. 13)
Geodesies, existence and uniqueness      328
Geodesies, frame fields and      250(Ex. 1)
Geodesies, length-minimizing properties      339—359
Geodesies, preserved by (local) isometries      275(Ex. 1) 326 362—363
Geodesies, spreading of      353—354
Geographical patch      134—135
Geometric surface      305 308
Gradient      32(Ex. 8) 50(Ex.
Gradient as normal vector field      148 216
Group      103
Group, euclidean      103
Group, Euclidean symmetry      302(Ex. 7)
Group, isometry      365
Group, orthogonal      104
Halmos symbol      9
Handle      379 382
Hausdorff axiom      186(Ex. 5) 345n
Helicoid      141(Ex. 7)
Helicoid as ruled minimal surface      227 233(Ex.
Helicoid, Gauss mapping      296(Ex. 20)
Helicoid, local isometries      267 275—276
Helicoid, patch computations for      213—214
Helix      15—16 58—59 119
Hilbert's Lemma      261
Hilbert's theorem      263
Holonomy      323—325
Holonomy angle      323
Holonomy, Gaussian curvature and      325(Ex. 5)
Homogeneous surface      366—368 370
Homotopic to constant      175
Hopf      386 389
Hyperbolic paraboloid      143(Ex. 12) 220(Ex.
Hyperbolic plane      315—316
Hyperbolic plane of pseudo-radius r      317(Ex. 4)
Hyperbolic plane, geodesic completeness      351(Ex. 1)
Hyperbolic plane, geodesies      334—336
Hyperbolic plane, isometries      371(Ex. 14)
Hyperbolic plane, local isometries      364
Identity mapping      99
Image      1
Image curve      33
Immersed surface      187(Ex. 10) 219 368
Improper integral      285—286
Induced inner product      308 313
Initial velocity      21(Ex. 6)
Inner product      304 316—317
Integral curve      186—187
Integral of function      286 292(Ex.
Integration of differential forms      167—176 283—297
Integration of differential forms, 1-forms over 1-segments      167—169 172—173
Integration of differential forms, 1-forms over oriented regions      285 292(Ex.
Integration of differential forms, 2-forms over 2-segments      169—172 174(Ex. 297(Ex.
Interior angle      374
Intrinsic distance      264—265 269(Ex. 351
Intrinsic geometry      271 304
Inverse function      2
Inverse function theorem      39 161—162
Isometric imbedding      367—369
Isometric immersion      367—369
Isometric invariant      271 304
Isometric surfaces      265
Isometry group      365
Isometry of Euclidean space      98—111
Isometry of Euclidean space, decomposition theorem      101
Isometry of Euclidean space, derivative map      104
Isometry of Euclidean space, determined by frames      105
Isometry of surfaces      263—266 270—275 306
Isometry of surfaces, Euclidean isometries and      297—299
Isometry of surfaces, isometric immersions and      367
Isothermal coordinates      279(Ex. 2)
J (rotation operator)      309(Ex. 5)
Jacobi equation      355—357 361(Ex.
Jacobian      288 294(Ex. 296(Ex.
Jacobian matrix      37
Kronecker delta      23 45
Lagrange identity      209(Ex. 6)
Law of Cosines      370(Ex. 10)
Leibnizian property      13
Length of a curve segment      see "Arc length"
Length of a vector      44
Levi Civita      322
Liebmann's Theorem      262
Line of curvature      see "Principal curve"
Liouville parametrization      339(Ex. 18)
Local isometry      265—270 362—365
Local isometry determined by frames      363
Local isometry of constant curvature surfaces      364
Local isometry, criteria for      266 269(Ex. 270(Ex.
Local minimization of arc length      352—358
Manifold      184—187
Mapping of Euclidean spaces      32—41
Mapping of surfaces      158—166
Mean curvature      203 205—208 212 217 252
Mercator projection      271(Ex. 13)
Metric tensor      305
Minimal surface      207 275(Ex.
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