Modern differential geometry of curves and surfaces:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Boca Raton [u.a.]
CRC Press
1993
|
| Schriftenreihe: | Studies in advanced mathematics
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005408758&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XVIII, 664 S. Ill., graph. Darst. |
| ISBN: | 0849378729 |
Internformat
MARC
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| 245 | 1 | 0 | |a Modern differential geometry of curves and surfaces |c Alfred Gray |
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| 300 | |a XVIII, 664 S. |b Ill., graph. Darst. | ||
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Datensatz im Suchindex
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| DE-BY-TUM_media_number | 040010335912 TEMP2935920 TEMP2935921 TEMP2935922 |
| _version_ | 1821941004799836162 |
| adam_text | CONTENTS
1. Curves in the Plane 1
1.1 Euclidean Spaces 2
1.2 Curves in Rn 4
1.3 The Length of a Curve 6
1.4 Vector Fields along Curves 10
1.5 Curvature of Curves in the Plane 11
1.6 The Turning Angle 13
1.7 The Semicubical Parabola 15
1.8 Exercises 16
2. Studying Curves in the Plane with Mathematica 17
2.1 Computing Curvature of Curves in the Plane 20
2.2 Computing Lengths of Curves 23
2.3 Filling Curves 24
2.4 Examples of Curves in R2 25
2.5 Plotting Piecewise Defined Curves 30
2.6 Exercises 33
ix
X
3. Famous Plane Curves 37
3.1 Cycloids 37
3.2 Lemniscates of Bernoulli 39
3.3 Cardioids 41
3.4 The Cissoid of Diodes 43
3.5 The Tractrix 46
3.6 Clothoids 50
3.7 Exercises 53
4. Alternate Methods for Plotting Plane Curves 57
4.1 Implicitly Defined Curves in M2 57
4.2 Cassinian Ovals 63
4.3 Plane Curves in Polar Coordinates 66
4.4 Exercises 71
5. New Curves from Old 75
5.1 Evolutes 76
5.2 Iterated Evolutes 79
5.3 The Evolute of a Tractrix is a Catenary 80
5.4 Involutes 81
5.5 Tangent and Normal Lines to Plane Curves 85
5.6 Osculating Circles to Plane Curves 90
5.7 Parallel Curves 95
5.8 Pedal Curves 97
5.9 Exercises 100
xj
6. Determining a Plane Curve from its Curvature 103
6.1 Euclidean Motions 104
6.2 Curves and Euclidean Motions 108
6.3 Intrinsic Equations for Plane Curves 109
6.4 Drawing Plane Curves with Assigned Curvature 113
6.5 Exercises 119
7. Curves in Space 123
7.1 Preliminaries 124
7.2 Curvature and Torsion of Unit Speed Curves in R3 125
7.3 Curvature and Torsion of Arbitrary Speed Curves in M3 129
7.4 Computing Curvature and Torsion with Mathematica 133
7.5 The Helix and its Generalizations 138
7.6 Viviani s Curve 140
7.7 The Fundamental Theorem of Space Curves 142
7.8 Drawing Space Curves with Assigned Curvature 145
7.9 Exercises 148
8. Tubes and Knots 153
8.1 Tubes about Curves 153
8.2 Torus Knots 155
8.3 Exercises 161
xiv
14.5 The Gaussian and Mean Curvatures 279
14.6 The Three Fundamental Forms 286
14.7 Examples of Curvature Calculations by Hand 287
14.8 The Curvature of Nonparametrically Defined Surfaces 291
14.9 Exercises 297
15. Surfaces in 3 Dimensional Space via Mathematica .... 299
15.1 Programs for Computing the Shape Operator and Curvature 299
15.2 Examples of Curvature Calculations with Mathematica 303
15.3 The Gauss Map via Mathematica 310
15.4 Exercises 316
16. Asymptotic Curves on Surfaces 319
16.1 Asymptotic Curves 320
16.2 Examples of Asymptotic Curves 323
16.3 Using Mathematica to Find Asymptotic Curves 328
16.4 Exercises 331
17. Ruled Surfaces 333
17.1 Examples of Ruled Surfaces 334
17.2 Flat Ruled Surfaces 341
17.3 Noncylindrical Ruled Surfaces 345
17.4 Examples of Striction Curves of Noncylindrical Ruled Surfaces.. 349
17.5 A Program for Ruled Surfaces 350
17.6 Developables 352
17.7 Exercises 354
XV
18. Surfaces of Revolution 357
18.1 Principal Curves 359
18.2 The Curvature of a Surface of Revolution 361
18.3 Generating a Surface of Revolution with Mathematics 365
18.4 The Catenoid 367
18.5 The Hyperboloid of Revolution 369
18.6 The Surfaces of Revolution of Curves with
Specified Curvature 370
18.7 Exercises 372
19. Surfaces of Constant Gaussian Curvature 375
19.1 The Elliptic Integral of the Second Kind 376
19.2 Surfaces of Revolution of Constant Positive Curvature 376
19.3 Surfaces of Revolution of Constant Negative Curvature 380
19.4 Kuen s Surface 384
19.5 Exercises 386
20. Intrinsic Surface Geometry 389
20.1 Intrinsic Formulas for the Gaussian Curvature 390
20.2 Gauss s Theorema Egregium 395
20.3 Christoffel Symbols 397
20.4 The Mainardi Codazzi Equations 401
20.5 Geodesic Curvature 402
20.6 Exercises 408
xvi .
21. Principal Curves and Umbilic Points 409
21.1 The Differential Equation for the
Principal Curves 410
21.2 Umbilic Points 413
21.3 Triply Orthogonal Systems of Surfaces 418
21.4 Elliptic Coordinates 424
21.5 Parabolic Coordinates 429
21.6 Exercises 432
22. Minimal Surfaces 1 435
22.1 Normal Variation 435
22.2 Examples of Minimal Surfaces 438
22.3 The Gauss Map of a Minimal Surface 449
22.4 Exercises 451
23. Minimal Surfaces II 455
23.1 Isothermal Coordinates 455
23.2 Minimal Surfaces and Complex Function Theory 456
23.3 Finding Conjugate Minimal Surfaces 462
23.4 Enneper s Surface of Degree n 469
23.5 The Weierstrass Representation 473
23.6 The Weierstrass Patches via Mathematica 476
23.7 Examples of Weierstrass Patches 477
23.8 Exercises 479
24. Construction of Surfaces 481
24.1 Parallel Surfaces 481
xyii
24.2 The Shape Operator of a Parallel Surface 485
24.3 Pedal Surfaces 488
24.4 Generalized Helicoids 489
24.5 Twisted Surfaces 495
24.6 Exercises 498
25. Differentiable Manifolds 499
25.1 The Definition of Differentiable Manifold 500
25.2 Differentiable Functions on Differentiable Manifolds 504
25.3 Tangent Vectors on Differentiable Manifolds 510
25.4 Induced Maps 518
25.5 Vector Fields on Differentiable Manifolds 524
25.6 Tensor Fields on Differentiable Manifolds 528
25.7 Exercises 532
26. Riemannian Manifolds 533
26.1 Covariant Derivatives 534
26.2 Indefinite Riemannian Metrics 540
26.3 The Classical Treatment of Metrics 544
27. Abstract Surfaces 549
27.1 Metrics on Abstract Surfaces 550
27.2 Examples of Abstract Surfaces 553
27.3 Computing Curvature of Metrics on Abstract Surfaces 555
27.4 Orientability of an Abstract Surface 557
27.5 Geodesic Curvature for Abstract Surfaces 558
27.6 Exercises 559
xviii
28. Geodesies on Surfaces 561
28.1 The Geodesic Equations 567
28.2 Clairaut Patches 564
28.3 Examples of Clairaut Patches 567
28.4 Finding Geodesies Numerically with Mathematica 569
28.5 Exercises 574
Appendix 575
A. 1 General Programs 575
A.2 Plane Curves 607
A.3 Space Curves 620
A.4 Surfaces 622
A.5 Metrics 636
A.6 Mathematica to Acrospin 636
Bibliography 645
Index 658
|
| any_adam_object | 1 |
| author | Gray, Alfred 1939-1998 |
| author_GND | (DE-588)124637108 |
| author_facet | Gray, Alfred 1939-1998 |
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| building | Verbundindex |
| bvnumber | BV008196650 |
| classification_rvk | SK 370 |
| classification_tum | MAT 533f MAT 532f |
| ctrlnum | (OCoLC)260205656 (DE-599)BVBBV008196650 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV008196650 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T09:18:12Z |
| institution | BVB |
| isbn | 0849378729 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005408758 |
| oclc_num | 260205656 |
| open_access_boolean | |
| owner | DE-384 DE-12 DE-355 DE-BY-UBR DE-824 DE-29T DE-91G DE-BY-TUM DE-91 DE-BY-TUM DE-739 DE-19 DE-BY-UBM DE-20 DE-634 DE-83 DE-11 DE-188 |
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| physical | XVIII, 664 S. Ill., graph. Darst. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Studies in advanced mathematics |
| spellingShingle | Gray, Alfred 1939-1998 Modern differential geometry of curves and surfaces Kurve (DE-588)4033824-1 gnd Differentialgeometrie (DE-588)4012248-7 gnd Oberfläche (DE-588)4042907-6 gnd Mathematica Programm (DE-588)4268208-3 gnd |
| subject_GND | (DE-588)4033824-1 (DE-588)4012248-7 (DE-588)4042907-6 (DE-588)4268208-3 |
| title | Modern differential geometry of curves and surfaces |
| title_auth | Modern differential geometry of curves and surfaces |
| title_exact_search | Modern differential geometry of curves and surfaces |
| title_full | Modern differential geometry of curves and surfaces Alfred Gray |
| title_fullStr | Modern differential geometry of curves and surfaces Alfred Gray |
| title_full_unstemmed | Modern differential geometry of curves and surfaces Alfred Gray |
| title_short | Modern differential geometry of curves and surfaces |
| title_sort | modern differential geometry of curves and surfaces |
| topic | Kurve (DE-588)4033824-1 gnd Differentialgeometrie (DE-588)4012248-7 gnd Oberfläche (DE-588)4042907-6 gnd Mathematica Programm (DE-588)4268208-3 gnd |
| topic_facet | Kurve Differentialgeometrie Oberfläche Mathematica Programm |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005408758&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT grayalfred moderndifferentialgeometryofcurvesandsurfaces |
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Teilbibliothek Mathematik & Informatik
| Signatur: |
0102 MAT 530f 2001 A 32796
Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |
Handapparate (nicht verfügbar)
| Signatur: |
0545 GRAY
Lageplan 0739 528 Lageplan 0740 MA.4179 Lageplan |
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| Exemplar 1 | Dauerhaft ausgeliehen Ausgeliehen |
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