Modern differential geometry of curves and surfaces with Mathematica:
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| Previous Title: | Gray, Alfred Modern differential geometry of curves and surfaces |
|---|---|
| Main Author: | |
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
1998
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIV, 1053 S.: Ill., graph. Darst. |
| ISBN: | 0849371643 |
Staff View
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| 245 | 1 | 0 | |a Modern differential geometry of curves and surfaces with Mathematica |c Alfred Gray |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 1998 | |
| 300 | |a XXIV, 1053 S.: Ill., graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804126273828552704 |
|---|---|
| adam_text | CONTENTS
1. Curves in the Plane 1
1.1 Euclidean Spaces 2
1.2 Curves in K 5
1.3 The Length of a Curve 7
1.4 Vector Fields along Curves 12
1.5 Curvature of Curves in the Plane 14
1.6 Angle Functions Between Plane Curves 17
1.7 The Turning Angle 19
1.8 The Semicubical Parabola 21
1.9 Exercises 22
2. Studying Curves in the Plane with Mathematica 25
2.1 Computing Curvature of Curves in the Plane 29
2.2 Computing Lengths of Curves 33
2.3 Filling Curves 34
2.4 Examples of Curves in K2 35
2.5 Plotting Piecewise Defined Curves 42
2.6 Exercises 45
xi
xjj
2.7 Animations of Definitions of Curves 47
3. Famous Plane Curves 49
3.1 Cycloids 50
3.2 Lemniscates of Bernoulli 52
3.3 Cardioids 54
3.4 The Catenary 55
3.5 The Cissoid of Diodes 57
3.6 The Tractrix 61
3.7 Clothoids 64
3.8 Pursuit Curves 66
3.9 Exercises 70
3.10 Animation of a Prolate Cycloid 74
4. Alternate Methods for Plotting Plane Curves 75
4.1 Implicitly Defined Curves in R2 75
4.2 Cassinian Ovals 82
4.3 Plane Curves in Polar Coordinates 86
4.4 Exercises 92
4.5 Animation of Fermat s Spiral 96
5. New Curves from Old 97
5.1 Evolutes 98
5.2 Iterated Evolutes 101
5.3 The Evolute of a Tractrix is a Catenary 702
5.4 Involutes 103
_____ xiji
5.5 Tangent and Normal Lines to Plane Curves 108
5.6 Osculating Circles to Plane Curves m
5.7 Parallel Curves 115
5.8 Pedal Curves /17
5.9 Exercises 120
5.10 Animation of a Cycloidial Pendulum 126
6. Determining a Plane Curve from its Curvature 727
6.1 Euclidean Motions 128
6.2 Curves and Euclidean Motions 134
6.3 Intrinsic Equations for Plane Curves 136
6.4 Drawing Plane Curves with Assigned Curvature 140
6.5 Exercises 146
6.6 Animation of Epicycloids and Hypocycloids 152
7. Global Properties of Plane Curves 153
7.1 The Total Signed Curvature of a Plane Curve 154
7.2 The Rotation Index of a Closed Plane Curve 159
7.3 Convex Plane Curves 163
7.4 The Four Vertex Theorem 165
7.5 Curves of Constant Width 168
7.6 Envelopes of Curves 172
7.7 The Support Function of an Oval 173
7.8 Reuleaux Polygons 176
7.9 The Involute Construction of Curves of Constant Width 177
7.10 Exercises 178
xiv
8. Curves in Space 181
8.1 The Vector Cross Product on E3 182
8.2 Curvature and Torsion of Unit Speed Curves in R3 183
8.3 Curvature and Torsion of Arbitrary Speed Curves in R3 189
8.4 Computing Curvature and Torsion with Mathematica 194
8.5 The Helix and its Generalizations 198
8.6 Viviani s Curve 201
8.7 Exercises 202
8.8 Animation of the Frenet Frame of Viviani s Curve 206
9. Tubes and Knots 207
9.1 Tubes about Curves 207
9.2 Torus Knots 209
9.3 Exercises 215
10. Construction of Space Curves 217
10.1 The Fundamental Theorem of Space Curves 217
10.2 Drawing Space Curves with Assigned Curvature 222
10.3 Contact 225
10.4 Space Curves that Lie on a Sphere 231
10.5 Curves of Constant Slope 234
10.6 Loxodromes on Spheres 238
10.7 Exercises 240
10.8 Animation of Curves on a Sphere 243
XV
11. Calculus on Euclidean Space 245
11.1 Tangent Vectors to Rn 246
11.2 Tangent Vectors as Directional Derivatives 248
11.3 Tangent Maps 250
11.4 Vector Fields on Rn 255
11.5 Derivatives of Vector Fields on Rn 258
11.6 Curves Revisited 265
11.7 Exercises 266
12. Surfaces in Euclidean Space 269
12.1 Patches in Rn 269
12.2 Patches in R3 277
12.3 The Local Gauss Map 279
12.4 The Definition of a Regular Surface in Rn 281
12.5 Tangent Vectors to Regular Surfaces in R 286
12.6 Surface Mappings 288
12.7 Level Surfaces in R3 291
12.8 Exercises 293
13. Examples of Surfaces 295
13.1 The Graph of a Function of Two Variables 296
13.2 The Ellipsoid 301
13.3 The Stereographic Ellipsoid 302
13.4 Tori 304
13.5 The Paraboloid 307
13.6 Sea Shells 308
xvi
13.7 Patches with Singularities 309
13.8 Implicit Plots of Surfaces 311
13.9 Exercises 312
14. Nonorientable Surfaces 317
14.1 Orientability of Surfaces 317
14.2 Nonorientable Surfaces Described by Identifications 322
14.3 The Mobius Strip 325
14.4 The Klein Bottle 327
14.5 A Different Klein Bottle 329
14.6 Realizations of the Real Projective Plane 330
14.7 Coloring Surfaces with Mathematica 335
14.8 Exercises 337
14.9 Animation of Steiner s Roman Surface 340
15. Metrics on Surfaces 341
15.1 The Intuitive Idea of Distance on a Surface 341
15.2 Isometries and Conformal Maps of Surfaces 346
15.3 The Intuitive Idea of Area on a Surface 351
15.4 Programs for Computing Metrics and Areas on a Surface 353
15.5 Examples of Metrics 354
15.6 Exercises 356
16. Surfaces in 3 Dimensional Space 359
16.1 The Shape Operator 360
16.2 Normal Curvature 363
xvii
16.3 Calculation of the Shape Operator ..¦ 367
16.4 The Eigenvalues of the Shape Operator 371
16.5 The Gaussian and Mean Curvatures 373
16.6 The Three Fundamental Forms 380
16.7 Examples of Curvature Calculations by Hand 382
16.8 A Global Curvature Theorem 386
16.9 Exercises 387
17. Surfaces in 3 DimensionalSpace via Mathematica ...391
17.1 Programs for Computing the Shape Operator and Curvature ... 392
17.2 Examples of Curvature Calculations with Mathematica 395
17.3 Principal Curvatures via Mathematica 402
17.4 The Gauss Map via Mathematica 403
17.5 The Curvature of Nonparametrically Defined Surfaces 409
17.6 Exercises 415
18. Asymptotic Curves on Surfaces 417
18.1 Asymptotic Curves 418
18.2 Examples of Asymptotic Curves 422
18.3 Using Mathematica to Find Asymptotic Curves 426
18.4 Exercises 429
19. Ruled Surfaces 431
19.1 Examples of Ruled Surfaces 432
19.2 Flat Ruled Surfaces 439
19.3 Tangent Developables 441
xviii
19.4 Noncylindrical Ruled Surfaces 445
19.5 Examples of Striction Curves of Noncylindrical Ruled Surfaces . 449
19.6 A Program for Ruled Surfaces 450
19.7 Other Examples of Ruled Surfaces 452
19.8 Exercises 454
20. Surfaces of Revolution 457
20.1 Principal Curves 459
20.2 The Curvature of a Surface of Revolution 461
20.3 Generating a Surface of Revolution with Mathematica 465
20.4 The Catenoid 467
20.5 The Hyperboloid of Revolution 470
20.6 Surfaces of Revolution of Curves with Specified Curvature 471
20.7 Surfaces of Revolution Generated by Data 472
20.8 Generalized Helicoids 475
20.9 Exercises 478
21. Surfaces of Constant Gaussian Curvature ... 457
21.1 The Elliptic Integral of the Second Kind 482
21.2 Surfaces of Revolution of Constant Positive Curvature 482
21.3 Surfaces of Revolution of Constant Negative Curvature 486
21.4 Flat Generalized Helicoids 490
21.5 Dini s Surface 493
21.6 Kuen s Surface 496
21.7 Exercises 497
jdx
22. Intrinsic Surface Geometry 501
22.1 Intrinsic Formulas for the Gaussian Curvature 502
22.2 Gauss s Theorema Egregium 507
22.3 Christoffel Symbols 509
22.4 Geodesic Curvature and Torsion 513
22.5 Exercises 519
23. Different/able Manifolds 521
23.1 The Definition of Differentiate Manifold 522
23.2 Differentiate Functions on Differentiate Manifolds 526
23.3 Tangent Vectors on Differentiate Manifolds 532
23.4 Induced Maps 540
23.5 Vector Fields on Differentiate Manifolds 545
23.6 Tensor Fields on Differentiate Manifolds 550
23.7 Exercises 554
24. Riemannian Manifolds 557
24.1 Covariant Derivatives 558
24.2 Indefinite Riemannian Metrics 564
24.3 The Classical Treatment of Metrics 567
24.4 Exercises 572
25. Abstract Surfaces 573
25.1 Metrics and Christoffel Symbols for Abstract Surfaces 575
25.2 Examples of Metrics on Abstract Surfaces 578
25.3 Computing Curvature of Metrics on Abstract Surfaces 580
XX
25.4 Orientability of an Abstract Surface 582
25.5 Geodesic Curvature for Abstract Surfaces 583
25.6 The Mean Curvature Vector Field 589
25.7 Exercises 591
26. Geodesies on Surfaces 595
26.1 The Geodesic Equations 596
26.2 Clairaut Patches 604
26.3 Finding Geodesies Numerically with Mathematica 613
26.4 The Exponential Map and the Gauss Lemma 617
26.5 Length Minimizing Properties of Geodesies 621
26.6 An Abstract Surface as a Metric Space 623
26.7 Exercises 625
27. The Gauss Bonnet Theorem 627
27.1 The Local Gauss Bonnet Theorem 628
27.2 Topology of Surfaces 634
27.3 The Global Gauss Bonnet Theorem 636
27.4 Applications of the Gauss Bonnet Theorem 638
27.5 Exercises 640
28. Principal Curves and Umbilic Points 641
28.1 The Differential Equation for the Principal Curves of a Surface .. 642
28.2 Umbilic Points 645
28.3 The Peterson Mainardi Codazzi Equations 649
28.4 Hubert s Lemma and Liebmann s Theorem 652
xxi
28.5 The Fundamental Theorem of Surfaces 654
28.6 Exercises 659
29. Triply Orthogonal Systems of Surfaces 661
29.1 Examples of Triply Orthogonal Systems 662
29.2 Curvilinear Patches and Dupin s Theorem 665
29.3 Elliptic Coordinates 669
29.4 Parabolic Coordinates 676
29.5 Exercises 678
30. Minimal Surfaces 681
30.1 Normal Variation 681
30.2 Examples of Minimal Surfaces 684
30.3 The Gauss Map of a Minimal Surface 694
30.4 Exercises 696
30.5 Animations of Minimal Surfaces 699
31. Minimal Surfaces and Complex Variables 701
31.1 Isothermal Coordinates 702
31.2 Isometric Deformations of Minimal Surfaces 705
31.3 Complex Derivatives 710
31.4 Elementary Complex Vector Algebra 714
31.5 Minimal Curves 716
31.6 Finding Conjugate Minimal Surfaces 721
31.7 Enneper s Surface of Degree n 728
31.8 Exercises 732
xxii
31.9 Animations of Minimal Surfaces 734
32. Minimal Surfaces via the Weierstrass Representation 735
32.1 The Weierstrass Representation 736
32.2 Weierstrass Patches via Mathematica 740
32.3 Examples of Weierstrass Patches 741
32.4 Minimal Surfaces with One Planar End 744
32.5 Costa s Minimal Surface 747
32.6 Exercises 760
33. Minimal Surfaces via Bjorling s Formula 761
33.1 Bjorling s Formula 761
33.2 Minimal Surfaces from Plane Curves 764
33.3 Examples of Minimal Surfaces Constructed from Plane Curves . 766
33.4 Exercises 772
34. Construction of Surfaces 773
34.1 Parallel Surfaces 773
34.2 Parallel Surfaces to a Mobius Strip 777
34.3 The Shape Operator of a Parallel Surface 779
34.4 A General Construction of a Triply Orthogonal System 782
34.5 Pedal Surfaces 784
34.6 Twisted Surfaces 786
34.7 Exercises 788
xxiii
35. Canal Surfaces and Cyclides of Dupin 759
35.1 Surfaces Whose Focal Sets are 2 Dimensional 791
35.2 Canal Surfaces 798
35.3 Cyclides of Dupin via Focal Sets 809
35.4 Exercises 817
36. Inversions of Curves and Surfaces 819
36.1 The Definition of Inversion 819
36.2 Inversion of Curves 822
36.3 Inversion of Surfaces 825
36.4 Cyclides of Dupin via Inversions 830
36.5 Liouville s Theorem 832
36.6 Exercises 834
Appendices 837
A General Programs 837
B Curves 891
Parametrically Defined Plane Curves 891
Implicitly Defined Plane Curves 916
Polar Defined Plane Curves 923
Parametrically Defined Space Curves 927
C Surfaces 935
Parametrically Defined Surfaces 935
Implicitly Defined Surfaces 969
Drum Plots 974
Minimal Curves 976
Surface Metrics 984
xxiv
D Plotting Programs 987
Miscellaneous Plotting Programs 987
Mathematica to Acrospin Programs 994
Mathematica to Geomview Programs 999
Bibliography 1005
Index 1021
Name Index 1041
Miniprogram and Mathematica Command Index 1045
|
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| dewey-full | 516.3/6 516.3/621 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
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| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV011733153 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:14:50Z |
| institution | BVB |
| isbn | 0849371643 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007914638 |
| oclc_num | 36949690 |
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| owner_facet | DE-91 DE-BY-TUM DE-703 DE-20 DE-739 DE-898 DE-BY-UBR DE-29T DE-521 DE-523 DE-634 DE-11 |
| physical | XXIV, 1053 S.: Ill., graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | CRC Press |
| record_format | marc |
| spelling | Gray, Alfred 1939-1998 Verfasser (DE-588)124637108 aut Modern differential geometry of curves and surfaces with Mathematica Alfred Gray 2. ed. Boca Raton [u.a.] CRC Press 1998 XXIV, 1053 S.: Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematica (Computer file) Geometria diferencial larpcal Géométrie différentielle - Informatique Processamento de dados larpcal Datenverarbeitung Geometry, Differential -- Data processing Mathematica Programm (DE-588)4268208-3 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Mathematica Programm (DE-588)4268208-3 s DE-604 Frühere Auflage Gray, Alfred Modern differential geometry of curves and surfaces HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007914638&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gray, Alfred 1939-1998 Modern differential geometry of curves and surfaces with Mathematica Mathematica (Computer file) Geometria diferencial larpcal Géométrie différentielle - Informatique Processamento de dados larpcal Datenverarbeitung Geometry, Differential -- Data processing Mathematica Programm (DE-588)4268208-3 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4268208-3 (DE-588)4012248-7 |
| title | Modern differential geometry of curves and surfaces with Mathematica |
| title_auth | Modern differential geometry of curves and surfaces with Mathematica |
| title_exact_search | Modern differential geometry of curves and surfaces with Mathematica |
| title_full | Modern differential geometry of curves and surfaces with Mathematica Alfred Gray |
| title_fullStr | Modern differential geometry of curves and surfaces with Mathematica Alfred Gray |
| title_full_unstemmed | Modern differential geometry of curves and surfaces with Mathematica Alfred Gray |
| title_old | Gray, Alfred Modern differential geometry of curves and surfaces |
| title_short | Modern differential geometry of curves and surfaces with Mathematica |
| title_sort | modern differential geometry of curves and surfaces with mathematica |
| topic | Mathematica (Computer file) Geometria diferencial larpcal Géométrie différentielle - Informatique Processamento de dados larpcal Datenverarbeitung Geometry, Differential -- Data processing Mathematica Programm (DE-588)4268208-3 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Mathematica (Computer file) Geometria diferencial Géométrie différentielle - Informatique Processamento de dados Datenverarbeitung Geometry, Differential -- Data processing Mathematica Programm Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007914638&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT grayalfred moderndifferentialgeometryofcurvesandsurfaceswithmathematica |