Modern differential geometry of curves and surfaces with mathematica:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton
CRC Press
2006
|
| Edition: | 3. ed. |
| Series: | Studies in advanced mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 984 Seiten Illustrationen, Diagramme |
| ISBN: | 9781584884484 |
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MARC
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Record in the Search Index
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| adam_text | THIRD EDITION MODERN DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES WITH
MATHEMATICS ALFRED GRAY, ELSA ABBENA, AND SIMON SALAMON CHAPMAN &
HALL/CRC TAYLOR & FRANCIS GROUP BOCA RATON LONDON NEW YORK CHAPMAN ST
HALL/CRC IS AN IMPRINT OF THE TAYLOR & FRANCIS CROUP, AN INFORMA
BUSINESS CHAPTER SCHEME 1 CURVES IN THE PLANE 1 2 FAMOUS PLANE CURVES 39
3 ALTERNATIVE WAYS OF PLOTTING CURVES 73 4 NEW CURVES FROM OLD 99 5
DETERMINING A PLANE CURVE FROM ITS CURVATURE 127 6 GLOBAL PROPERTIES OF
PLANE CURVES 153 7 CURVES IN SPACE 191 8 CONSTRUCTION OF SPACE CURVES
229 9 CALCULUS ON EUCLIDEAN SPACE 263 10 SURFACES IN EUCLIDEAN SPACE 287
11 NONORIENTABLE SURFACES 331 12 METRICS ON SURFACES 361 13 SHAPE AND
CURVATURE 385 14 RULED SURFACES 431 15 SURFACES OF REVOLUTION AND
CONSTANT CURVATURE 461 16 A SELECTION OF MINIMAL SURFACES 501 17
INTRINSIC SURFACE GEOMETRY 531 18 ASYMPTOTIC CURVES AND GEODESIES ON
SURFACES 557 19 PRINCIPAL CURVES AND UMBILIC POINTS .. R. 593 20 CANAL
SURFACES AND CYCLIDES OF DUPIN 639 21 THE THEORY OF SURFACES OF CONSTANT
NEGATIVE CURVATURE 683 22 MINIMAL SURFACES VIA COMPLEX VARIABLES 719 23
ROTATION AND ANIMATION USING QUATERNIONS 767 24 DIFFERENTIABLE MANIFOLDS
809 25 RIEMANNIAN MANIFOLDS 847 26 ABSTRACT SURFACES AND THEIR GEODESIES
871 27 THE GAUSS-BONNET THEOREM 901 FULL CONTENTS 1. CURVES IN THE PLANE
1 1.1 EUCLIDEAN SPACE 2 1.2 CURVES IN SPACE 5 1.3 THE LENGTH OF A CURVE
9 1.4 CURVATURE OF PLANE CURVES 14 1.5 ANGLE FUNCTIONS 17 1.6 FIRST
EXAMPLES OF PLANE CURVES 20 1.7 THE SEMICUBICAL PARABOLA AND REGULARITY
25 1.8 EXERCISES 27 NOTEBOOK 1 29 2. FAMOUS PLANE CURVES 39 2.1 CYCLOIDS
40 2.2 LEMNISCATES OF BERNOULLI 43 2.3 CARDIOIDS 45 2.4 THE CATENARY 45
2.5 THE CISSOID OF DIODES 47 2.6 THE TRACTRIX 50 2.7 CLOTHOIDS 53 2.8
PURSUIT CURVES 54 2.9 EXERCISES 57 NOTEBOOK 2 . ! 60 3. ALTERNATIVE WAYS
OF PLOTTING CURVES 73 3.1 IMPLICITLY DEFINED PLANE CURVES 74 3.2 THE
FOLIUM OF DESCARTES 76 3.3 CASSINIAN OVALS 79 3.4 PLANE CURVES IN POLAR
COORDINATES 81 3.5 A SELECTION OF SPIRALS 82 3.6 EXERCISES 85 NOTEBOOK 3
88 4. NEW CURVES FROM OLD 99 4.1 EVOLUTES 99 4.2 ITERATED EVOLUTES 102
4.3 INVOLUTES 104 4.4 OSCULATING CIRCLES TO PLANE CURVES 107 4.5
PARALLEL CURVES 110 4.6 PEDAL CURVES 113 4.7 EXERCISES 115 NOTEBOOK 4
118 5. DETERMINING A PLANE CURVE FROM ITS CURVATURE 127 5.1 EUCLIDEAN
MOTIONS 128 5.2 ISOMETRIES OF THE PLANE 133 5.3 INTRINSIC EQUATIONS FOR
PLANE CURVES 136 5.4 EXAMPLES OF CURVES WITH ASSIGNED CURVATURE 140 5.5
EXERCISES 142 NOTEBOOK 5 146 6. GLOBAL PROPERTIES OF PLANE CURVES 153
6.1 TOTAL SIGNED CURVATURE , 154 6.2 TROCHOID CURVES 157 6.3 THE
ROTATION INDEX OF A CLOSED CURVE 160 6.4 CONVEX PLANE CURVES 164 6.5 THE
FOUR VERTEX THEOREM 166 6.6 CURVES OF CONSTANT WIDTH 169 6.7 REULEAUX
POLYGONS AND INVOLUTES 172 6.8 THE SUPPORT FUNCTION OF AN OVAL 174 6.9
EXERCISES 178 NOTEBOOK 6 181 7. CURVES IN SPACE 191 7.1 THE VECTOR CROSS
PRODUCT 192 7.2 CURVATURE AND TORSION OF UNIT-SPEED CURVES 195 7.3 THE
HELIX AND TWISTED CUBIC 200 7.4 ARBITRARY-SPEED CURVES IN M 3 203 7.5
MORE CONSTRUCTIONS OF SPACE CURVES 206 7.6 TUBES AND TORI 209 7.7 TORUS
KNOTS 211 7.8 EXERCISES 213 NOTEBOOK 7 217 8. CONSTRUCTION OF SPACE
CURVES 229 8.1 THE FUNDAMENTAL THEOREM OF SPACE CURVES 230 8.2 ASSIGNED
CURVATURE AND TORSION 233 8.3 CONTACT 236 8.4 SPACE CURVES THAT LIE ON A
SPHERE 242 8.5 CURVES OF CONSTANT SLOPE . 246 8.6 LOXODROMES ON SPHERES
250 8.7 EXERCISES 252 NOTEBOOK 8 254 9. CALCULUS ON EUCLIDEAN SPACE 263
9.1 TANGENT VECTORS TO R N 263 9.2 TANGENT VECTORS AS DIRECTIONAL
DERIVATIVES 265 9.3 TANGENT MAPS OR DIFFERENTIALS 268 9.4 VECTOR FIELDS
ON R* 272 9.5 DERIVATIVES OF VECTOR FIELDS 275 9.6 CURVES REVISITED 280
9.7 EXERCISES 281 NOTEBOOK 9 283 10. SURFACES IN EUCLIDEAN SPACE 287
10.1 PATCHES IN K 288 10.2 PATCHES IN R 3 AND THE LOCAL GAUSS MAP 295
10.3 THE DEFINITION OF A REGULAR SURFACE 297 10.4 EXAMPLES OF SURFACES
302 10.5 TANGENT VECTORS AND SURFACE MAPPINGS 307 10.6 LEVEL SURFACES IN
M 3 311 10.7 EXERCISES 316 NOTEBOOK 10 320 11. NONORIENTABLE SURFACES
331 11.1 ORIENTABILITY OF SURFACES 331 11.2 SURFACES BY IDENTIFICATION
336 11.3 THE MOBIUS STRIP 339 11.4 THE KLEIN BOTTLE 341 11.5
REALIZATIONS OF THE REAL PROJECTIVE PLANE 343 11.6 TWISTED SURFACES 348
11.7 EXERCISES 350 NOTEBOOK 11 352 12. METRICS ON SURFACES 361 12.1 THE
INTUITIVE IDEA OF DISTANCE 361 12.2 ISOMETRIES BETWEEN SURFACES 365 12.3
DISTANCE AND CONFORMAL MAPS 369 12.4 THE INTUITIVE IDEA OF AREA 372 12.5
EXAMPLES OF METRICS 374 12.6 EXERCISES 377 NOTEBOOK 12 379 13. SHAPE AND
CURVATURE 385 13.1 THE SHAPE OPERATOR 386 13.2 NORMAL CURVATURE 389 13.3
CALCULATION OF THE SHAPE OPERATOR 393 13.4 GAUSSIAN AND MEAN CURVATURE
397 13.5 MORE CURVATURE CALCULATIONS 403 13.6 A GLOBAL CURVATURE THEOREM
410 13.7 NONPARAMETRICALLY DEFINED SURFACES 411 13.8 EXERCISES 415
NOTEBOOK 13 420 14. RULED SURFACES 431 14.1 DEFINITIONS AND EXAMPLES 432
14.2 CURVATURE OF A RULED SURFACE 437 14.3 TANGENT DEVELOPABLES 440 14.4
NONCYLINDRICAL RULED SURFACES 444 14.5 EXERCISES 449 NOTEBOOK 14 .452
15. SURFACES OF REVOLUTION AND*CONSTANT CURVATURE ... 461 15.1 SURFACES
OF REVOLUTION 462 15.2 PRINCIPAL CURVES 465 15.3 CURVATURE OF A SURFACE
OF REVOLUTION 468 15.4 GENERALIZED HELICOIDS 470 15.5 SURFACES OF
CONSTANT POSITIVE CURVATURE 473 15.6 SURFACES OF CONSTANT NEGATIVE
CURVATURE 477 15.7 MORE EXAMPLES OF CONSTANT CURVATURE 480 15.8
EXERCISES 485 NOTEBOOK 15 488 16. A SELECTION OF MINIMAL SURFACES 501
16.1 NORMAL VARIATION 502 16.2 DEFORMATION FROM THE HELICOID TO THE
CATENOID 504 16.3 MINIMAL SURFACES OF REVOLUTION 507 16.4 MORE EXAMPLES
OF MINIMAL SURFACES 509 16.5 MONGE PATCHES AND SCHERK S-MINIMAL SURFACE
513 16.6 THE GAUSS MAP OF A MINIMAL SURFACE 516 16.7 ISOTHERMAL
COORDINATES 518 16.8 EXERCISES 521 NOTEBOOK 16 523 17. INTRINSIC SURFACE
GEOMETRY 531 17.1 INTRINSIC FORMULAS FOR THE GAUSSIAN CURVATURE 531 17.2
GAUSS S THEOREMA EGREGIUM 535 17.3 CHRISTOFFEL SYMBOLS 537 17.4 GEODESIC
CURVATURE OF CURVES ON SURFACES 540 17.5 GEODESIC TORSION AND FRENET
FORMULAS 545 17.6 EXERCISES 547 NOTEBOOK 17 548 18. ASYMPTOTIC CURVES
AND GEODESIES ON SURFACES ... 557 18.1 ASYMPTOTIC CURVES 558 18.2
EXAMPLES OF ASYMPTOTIC CURVES AND PATCHES 562 18.3 THE GEODESIC
EQUATIONS 565 18.4 FIRST EXAMPLES OF GEODESIES 569 18.5 CLAIRAUT PATCHES
572 18.6 USE OF CLAIRAUT PATCHES 576 18.7 EXERCISES 579 NOTEBOOK 18 582
19. PRINCIPAL CURVES AND UMBILIC POINTS 593 19.1 THE DIFFERENTIAL
EQUATION FOR PRINCIPAL CURVES 594 19.2 UMBILIC POINTS 597 19.3 THE
PETERSON-MAINARDI-CODAZZI EQUATIONS 599 19.4 HILBERT S LEMMA AND
LIEBMANN S THEOREM 602 19.5 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 604
19.6 ELLIPTIC COORDINATES 609 19.7 PARABOLIC COORDINATES AND A GENERAL
CONSTRUCTION 615 19.8 PARALLEL SURFACES 620 19.9 THE SHAPE OPERATOR OF A
PARALLEL SURFACE 622 19.10 EXERCISES 625 NOTEBOOK 19 628 20. CANAL
SURFACES AND CYCLIDES OF DUPIN 639 20.1 SURFACES WHOSE FOCAL SETS ARE
2-DIMENSIONAL 641 20.2 CANAL SURFACES 646 20.3 CYCLIDES OF DUPIN VIA
FOCAL SETS 655 20.4 THE DEFINITION OF INVERSION 661 20.5 INVERSION OF
SURFACES 665 20.6 EXERCISES 669 NOTEBOOK 20 673 21. THE THEORY OF
SURFACES OF CONSTANT NEGATIVE CURVATURE 683 21.1 INTRINSIC TCHEBYSHEF
PATCHES 684 21.2 PATCHES ON SURFACES OF CONSTANT NEGATIVE CURVATURE 687
21.3 THE SINE-GORDON EQUATION 691 21.4 TCHEBYSHEF PATCHES ON SURFACES OF
REVOLUTION 692 21.5 THE BIANCHI TRANSFORM 697 21.6 MOVING FRAMES ON
SURFACES IN M 3 701 21.7 KUEN S SURFACE AS BIANCHI TRANSFORM OF THE
PSEUDOSPHERE . 704 21.8 THE BACKLUND TRANSFORM 706 21.9 EXERCISES 710
NOTEBOOK 21 712 22. MINIMAL SURFACES VIA COMPLEX VARIABLES 719 22A
ISOMETRIC DEFORMATIONS OF MINIMAL SURFACES 720 22.2 COMPLEX DERIVATIVES
725 22.3 MINIMAL CURVES 728 22.4 FINDING CONJUGATE MINIMAL SURFACES 732
22.5 THE WEIERSTRASS REPRESENTATION 736 22.6 MINIMAL SURFACES VIA
BJORLING S FORMULA 741 22.7 COSTA S MINIMAL SURFACE 746 22.8 EXERCISES
752 NOTEBOOK 22 755 23. ROTATION AND ANIMATION USING QUATERNIONS 767
23.1 ORTHOGONAL MATRICES 769 23.2 QUATERNION ALGEBRA 775 23.3 UNIT
QUATERNIONS AND ROTATIONS 779 23.4 IMAGINARY QUATERNIONS AND ROTATIONS
783 23.5 ROTATION CURVES 785 23.6 EULER ANGLES : 789 23.7 FURTHER TOPICS
793 23.8 EXERCISES 797 NOTEBOOK 23 799 24. DIFFERENTIABLE MANIFOLDS 809
24.1 THE DEFINITION OF A DIFFERENTIABLE MANIFOLD 810 24.2 DIFFERENTIABLE
FUNCTIONS ON MANIFOLDS 814 24.3 TANGENT VECTORS ON MANIFOLDS 819 24.4
INDUCED MAPS 826 24.5 VECTOR FIELDS ON MANIFOLDS 831 24.6 TENSOR FIELDS
836 24.7 EXERCISES 839 NOTEBOOK 24 842 25. RIEMANNIAN MANIFOLDS 847 25.1
COVARIANT DERIVATIVES 848 25.2 PSEUDO-RIEMANNIAN METRICS 854 25.3 THE
CLASSICAL TREATMENT OF METRICS 858 25.4 THE CHRISTOFFEL SYMBOLS IN
RIEMANNIAN GEOMETRY 861 25.5 THE RIEMANN CURVATURE TENSOR 863 25.6
EXERCISES 867 NOTEBOOK 25 868 26. ABSTRACT SURFACES AND THEIR GEODESIES
871 26.1 CHRISTOFFEL SYMBOLS ON ABSTRACT SURFACES 872 26.2 EXAMPLES OF
ABSTRACT METRICS 874 26.3 THE ABSTRACT DEFINITION OF GEODESIC CURVATURE
878 26.4 GEODESIES ON ABSTRACT SURFACES 880 26.5 THE EXPONENTIAL MAP AND
THE GAUSS LEMMA 884 26.6 LENGTH MINIMIZING PROPERTIES OF GEODESIES 888
26.7 EXERCISES 892 NOTEBOOK 26 895 27. THE GAUSS-BONNET THEOREM 901 27.1
TURNING ANGLES AND LIOUVILLE S THEOREM 902 27.2 THE LOCAL GAUSS-BONNET
THEOREM 906 27.3 AN AREA BOUND 910 27.4 A GENERALIZATION TO MORE
COMPLICATED REGIONS 911 27.5 THE TOPOLOGY OF SURFACES 914 27.6 THE
GLOBAL GAUSS-BONNET THEOREM 918 27.7 APPLICATIONS OF THE GAUSS-BONNET
THEOREM 920 27.8 EXERCISES 921 NOTEBOOK 27 923 BIBLIOGRAPHY 931 NAME
INDEX 953 SUBJECT INDEX 957 NOTEBOOK INDEX 977
|
| any_adam_object | 1 |
| author | Gray, Alfred 1939-1998 Abbena, Elas Salamon, Simon |
| author_GND | (DE-588)124637108 (DE-588)120582555X (DE-588)1138726966 |
| author_facet | Gray, Alfred 1939-1998 Abbena, Elas Salamon, Simon |
| author_role | aut aut aut |
| author_sort | Gray, Alfred 1939-1998 |
| author_variant | a g ag e a ea s s ss |
| building | Verbundindex |
| bvnumber | BV024611991 |
| classification_rvk | SK 370 ST 601 |
| ctrlnum | (OCoLC)916641093 (DE-599)BVBBV024611991 |
| discipline | Informatik Mathematik |
| edition | 3. ed. |
| format | Book |
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| id | DE-604.BV024611991 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:03:01Z |
| institution | BVB |
| isbn | 9781584884484 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018584632 |
| oclc_num | 916641093 |
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| owner_facet | DE-83 DE-11 DE-355 DE-BY-UBR DE-473 DE-BY-UBG |
| physical | 984 Seiten Illustrationen, Diagramme |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Studies in advanced mathematics |
| spelling | Gray, Alfred 1939-1998 Verfasser (DE-588)124637108 aut Modern differential geometry of curves and surfaces with mathematica Alfred Gray, Elsa Abbena, and Simon Salamon 3. ed. Boca Raton CRC Press 2006 984 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Studies in advanced mathematics 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 Abbena, Elas Verfasser (DE-588)120582555X aut Salamon, Simon Verfasser (DE-588)1138726966 aut Erscheint auch als Onlineausgabe 978-1-315-27603-8 Erscheint auch als Onlineausgabe 978-1-351-99220-6 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018584632&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gray, Alfred 1939-1998 Abbena, Elas Salamon, Simon Modern differential geometry of curves and surfaces with mathematica 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, Elsa Abbena, and Simon Salamon |
| title_fullStr | Modern differential geometry of curves and surfaces with mathematica Alfred Gray, Elsa Abbena, and Simon Salamon |
| title_full_unstemmed | Modern differential geometry of curves and surfaces with mathematica Alfred Gray, Elsa Abbena, and Simon Salamon |
| title_short | Modern differential geometry of curves and surfaces with mathematica |
| title_sort | modern differential geometry of curves and surfaces with mathematica |
| topic | Mathematica Programm (DE-588)4268208-3 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Mathematica Programm Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018584632&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT grayalfred moderndifferentialgeometryofcurvesandsurfaceswithmathematica AT abbenaelas moderndifferentialgeometryofcurvesandsurfaceswithmathematica AT salamonsimon moderndifferentialgeometryofcurvesandsurfaceswithmathematica |