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Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition 2nd New edition [Kõva köide]

(American University, Washington, District of Columbia, USA)
  • Formaat: Hardback, 1088 pages, kõrgus x laius: 254x178 mm, kaal: 2109 g, 71 Halftones, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 29-Dec-1997
  • Kirjastus: CRC Press Inc
  • ISBN-10: 0849371643
  • ISBN-13: 9780849371646
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Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition 2nd New editionSuurem pilt
  • Formaat: Hardback, 1088 pages, kõrgus x laius: 254x178 mm, kaal: 2109 g, 71 Halftones, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 29-Dec-1997
  • Kirjastus: CRC Press Inc
  • ISBN-10: 0849371643
  • ISBN-13: 9780849371646
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780849371646.html
Märksõnad:
This text for a junior-senior level course in differential geometry presents standard material about curves and surfaces along with instructions for using the symbolic manipulation program Mathematica for making pictures and computing such functions as curvature and torsion. The text is compatible with versions 2.2 and 3.0, and can serve as an introduction to the program. Appendices include numerous general Mathematica programs, and programs for curves, surfaces, and plotting. Annotation c. by Book News, Inc., Portland, Or.

The Second Edition combines a traditional approach with the symbolic manipulation abilities of Mathematica to explain and develop the classical theory of curves and surfaces. You will learn to reproduce and study interesting curves and surfaces - many more than are included in typical texts - using computer methods. By plotting geometric objects and studying the printed result, teachers and students can understand concepts geometrically and see the effect of changes in parameters.

Modern Differential Geometry of Curves and Surfaces with Mathematica explains how to define and compute standard geometric functions, for example the curvature of curves, and presents a dialect of Mathematica for constructing new curves and surfaces from old. The book also explores how to apply techniques from analysis.

Although the book makes extensive use of Mathematica, readers without access to that program can perform the calculations in the text by hand. While single- and multi-variable calculus, some linear algebra, and a few concepts of point set topology are needed to understand the theory, no computer or Mathematica skills are required to understand the concepts presented in the text. In fact, it serves as an excellent introduction to Mathematica, and includes fully documented programs written for use with Mathematica.
Ideal for both classroom use and self-study, Modern Differential Geometry of Curves and Surfaces with Mathematica has been tested extensively in the classroom and used in professional short courses throughout the world.
1. Curves in the Plane
1(24)
1.1 Euclidean Spaces
2(3)
1.2 Curves in R(n)
5(2)
1.3 The Length of a Curve
7(5)
1.4 Vector Fields along Curves
12(2)
1.5 Curvature of Curves in the Plane
14(3)
1.6 Angle Functions Between Plane Curves
17(2)
1.7 The Turning Angle
19(2)
1.8 The Semicubical Parabola
21(1)
1.9 Exercises
22(3)
2. Studying Curves in the Plane with Mathematica
25(24)
2.1 Computing Curvature of Curves in the Plane
29(4)
2.2 Computing Lengths of Curves
33(1)
2.3 Filling Curves
34(1)
2.4 Examples of Curves in R(2)
35(7)
2.5 Plotting Piecewise Defined Curves
42(3)
2.6 Exercises
45(2)
2.7 Animations of Definitions of Curves
47(2)
3. Famous Plane Curves
49(26)
3.1 Cycloids
50(2)
3.2 Lemniscates of Bernoulli
52(2)
3.3 Cardioids
54(1)
3.4 The Catenary
55(2)
3.5 The Cissoid of Diocles
57(4)
3.6 The Tractrix
61(3)
3.7 Clothoids
64(2)
3.8 Pursuit Curves
66(4)
3.9 Exercises
70(4)
3.10 Animation of a Prolate Cycloid
74(1)
4. Alternate Methods for Plotting Plane Curves
75(22)
4.1 Implicitly Defined Curves in R(2)
75(7)
4.2 Cassinian Ovals
82(4)
4.3 Plane Curves in Polar Coordinates
86(6)
4.4 Exercises
92(4)
4.5 Animation of Fermat's Spiral
96(1)
5. New Curves from Old
97(30)
5.1 Evolutes
98(3)
5.2 Iterated Evolutes
101(1)
5.3 The Evolute of a Tractrix is a Catenary
102(1)
5.4 Involutes
103(5)
5.5 Tangent and Normal Lines to Plane Curves
108(3)
5.6 Osculating Circles to Plane Curves
111(4)
5.7 Parallel Curves
115(2)
5.8 Pedal Curves
117(3)
5.9 Exercises
120(6)
5.10 Animation of a Cycloidial Pendulum
126(1)
6. Determining a Plane Curve from its Curvature
127(26)
6.1 Euclidean Motions
128(6)
6.2 Curves and Euclidean Motions
134(2)
6.3 Intrinsic Equations for Plane Curves
136(4)
6.4 Drawing Plane Curves with Assigned Curvature
140(6)
6.5 Exercises
146(6)
6.6 Animation of Epicycloids and Hypocycloids
152(1)
7. Global Properties of Plane Curves
153(28)
7.1 The Total Signed Curvature of a Plane Curve
154(5)
7.2 The Rotation Index of a Closed Plane Curve
159(4)
7.3 Convex Plane Curves
163(2)
7.4 The Four Vertex Theorem
165(3)
7.5 Curves of Constant Width
168(4)
7.6 Envelopes of Curves
172(1)
7.7 The Support Function of an Oval
173(3)
7.8 Reuleaux Polygons
176(1)
7.9 The Involute Construction of Curves of Constant Width
177(1)
7.10 Exercises
178(3)
8. Curves in Space
181(26)
8.1 The Vector Cross Product on R(3)
182(1)
8.2 Curvature and Torsion of Unit-Speed Curves in R(3)
183(6)
8.3 Curvature and Torsion of Arbitrary-Speed Curves in R(3)
189(5)
8.4 Computing Curvature and Torsion with Mathematica
194(4)
8.5 The Helix and its Generalizations
198(3)
8.6 Viviani's Curve
201(1)
8.7 Exercises
202(4)
8.8 Animation of the Frenet Frame of Viviani's Curve
206(1)
9. Tubes and Knots
207(10)
9.1 Tubes about Curves
207(2)
9.2 Torus Knots
209(6)
9.3 Exercises
215(2)
10. Construction of Space Curves
217(28)
10.1 The Fundamental Theorem of Space Curves
217(5)
10.2 Drawing Space Curves with Assigned Curvature
222(3)
10.3 Contact
225(6)
10.4 Space Curves that Lie on a Sphere
231(3)
10.5 Curves of Constant Slope
234(4)
10.6 Loxodromes on Spheres
238(2)
10.7 Exercises
240(3)
10.8 Animation of Curves on a Sphere
243(2)
11. Calculus on Euclidean Space
245(24)
11.1 Tangent Vectors to R(n)
246(2)
11.2 Tangent Vectors as Directional Derivatives
248(2)
11.3 Tangent Maps
250(5)
11.4 Vector Fields on R(n)
255(3)
11.5 Derivatives of Vector Fields on R(n)
258(7)
11.6 Curves Revisited
265(1)
11.7 Exercises
266(3)
12. Surfaces in Euclidean Space
269(26)
12.1 Patches in R(n)
269(8)
12.2 Patches in R(3)
277(2)
12.3 The Local Gauss Map
279(2)
12.4 The Definition of a Regular Surface in R(n)
281(5)
12.5 Tangent Vectors to Regular Surfaces in R(n)
286(2)
12.6 Surface Mappings
288(3)
12.7 Level Surfaces in R(3)
291(2)
12.8 Exercises
293(2)
13. Examples of Surfaces
295(22)
13.1 The Graph of a Function of Two Variables
296(5)
13.2 The Ellipsoid
301(1)
13.3 The Stereographic Ellipsoid
302(2)
13.4 Tori
304(3)
13.5 The Paraboloid
307(1)
13.6 Sea Shells
308(1)
13.7 Patches with Singularities
309(2)
13.8 Implicit Plots of Surfaces
311(1)
13.9 Exercises
312(5)
14. Nonorientable Surfaces
317(24)
14.1 Orientability of Surfaces
317(5)
14.2 Nonorientable Surfaces Described by Identifications
322(3)
14.3 The Mobius Strip
325(2)
14.4 The Klein Bottle
327(2)
14.5 A Different Klein Bottle
329(1)
14.6 Realizations of the Real Projective Plane
330(5)
14.7 Coloring Surfaces with Mathematica
335(2)
14.8 Exercises
337(3)
14.9 Animation of Steiner's Roman Surface
340(1)
15. Metrics on Surfaces
341(18)
15.1 The Intuitive Idea of Distance on a Surface
341(5)
15.2 Isometries and Conformal Maps of Surfaces
346(5)
15.3 The Intuitive Idea of Area on a Surface
351(2)
15.4 Programs for Computing Metrics and Area on a Surface
353(1)
15.5 Examples of Metrics
354(2)
15.6 Exercises
356(3)
16. Surfaces in 3-Dimensional Space
359(32)
16.1 The Shape Operator
360(3)
16.2 Normal Curvature
363(4)
16.3 Calculation of the Shape Operator
367(4)
16.4 The Eigenvalues of the Shape Operator
371(2)
16.5 The Gaussian and Mean Curvatures
373(7)
16.6 The Three Fundamental Forms
380(2)
16.7 Examples of Curvature Calculations by Hand
382(4)
16.8 A Global Curvature Theorem
386(1)
16.9 Exercises
387(4)
17. Surfaces in 3-Dimensional Space via Mathematica
391(26)
17.1 Programs for Computing the Shape Operator and Curvature
392(3)
17.2 Examples of Curvature Calculations with Mathematica
395(7)
17.3 Principal Curvatures via Mathematica
402(1)
17.4 The Gauss Map via Mathematica
403(6)
17.5 The Curvature of Nonparametrically Defined Surfaces
409(6)
17.6 Exercises
415(2)
18. Asymptotic Curves on Surfaces
417(14)
18.1 Asymptotic Curves
418(4)
18.2 Examples of Asymptotic Curves
422(4)
18.3 Using Mathematica to Find Asymptotic Curves
426(3)
18.4 Exercises
429(2)
19. Ruled Surfaces
431(26)
19.1 Examples of Ruled Surfaces
432(7)
19.2 Flat Ruled Surfaces
439(2)
19.3 Tangent Developables
441(4)
19.4 Noncylindrical Ruled Surfaces
445(4)
19.5 Examples of Striction Curves of Noncylindrical Ruled Surfaces
449(1)
19.6 A Program for Ruled Surfaces
450(2)
19.7 Other Examples of Ruled Surfaces
452(2)
19.8 Exercises
454(3)
20. Surfaces of Revolution
457(24)
20.1 Principal Curves
459(2)
20.2 The Curvature of a Surface of Revolution
461(4)
20.3 Generating a Surface of Revolution with Mathematica
465(2)
20.4 The Catenoid
467(3)
20.5 The Hyperboloid of Revolution
470(1)
20.6 Surfaces of Revolution of Curves with Specified Curvature
471(1)
20.7 Surfaces of Revolution Generated by Data
472(3)
20.8 Generalized Helicoids
475(3)
20.9 Exercises
478(3)
21. Surfaces of Constant Gaussian Curvature
481(20)
21.1 The Elliptic Integral of the Second Kind
482(1)
21.2 Surfaces of Revolution of Constant Positive Curvature
482(4)
21.3 Surfaces of Revolution of Constant Negative Curvature
486(4)
21.4 Flat Generalized Helicoids
490(3)
21.5 Dini's Surface
493(3)
21.6 Kuen's Surface
496(1)
21.7 Exercises
497(4)
22. Intrinsic Surface Geometry
501(20)
22.1 Intrinsic Formulas for the Gaussian Curvature
502(5)
22.2 Gauss's Theorema Egregium
507(2)
22.3 Christoffel Symbols
509(4)
22.4 Geodesic Curvature and Torsion
513(6)
22.5 Exercises
519(2)
23. Differentiable Manifolds
521(36)
23.1 The Definition of Differentiable Manifold
522(4)
23.2 Differentiable Functions on Differentiable Manifolds
526(6)
23.3 Tangent Vectors on Differentiable Manifolds
532(8)
23.4 Induced Maps
540(5)
23.5 Vector Fields on Differentiable Manifolds
545(5)
23.6 Tensor Fields on Differentiable Manifolds
550(4)
23.7 Exercises
554(3)
24. Riemannian Manifolds
557(16)
24.1 Covariant Derivatives
558(6)
24.2 Indefinite Riemannian Metrics
564(3)
24.3 The Classical Treatment of Metrics
567(5)
24.4 Exercises
572(1)
25. Abstract Surfaces
573(22)
25.1 Metrics and Christoffel Symbols for Abstract Surfaces
575(3)
25.2 Examples of Metrics on Abstract Surfaces
578(2)
25.3 Computing Curvature of Metrics on Abstract Surfaces
580(2)
25.4 Orientability of an Abstract Surface
582(1)
25.5 Geodesic Curvature for Abstract Surfaces
583(6)
25.6 The Mean Curvature Vector Field
589(2)
25.7 Exercises
591(4)
26. Geodesics on Surfaces
595(32)
26.1 The Geodesic Equations
596(8)
26.2 Clairaut Patches
604(9)
26.3 Finding Geodesics Numerically with Mathematica
613(4)
26.4 The Exponential Map and the Gauss Lemma
617(4)
26.5 Length Minimizing Properties of Geodesics
621(2)
26.6 An Abstract Surface as a Metric Space
623(2)
26.7 Exercises
625(2)
27. The Gauss-Bonnet Theorem
627(14)
27.1 The Local Gauss-Bonnet Theorem
628(6)
27.2 Topology of Surfaces
634(2)
27.3 The Global Gauss-Bonnet Theorem
636(2)
27.4 Applications of the Gauss-Bonnet Theorem
638(2)
27.5 Exercises
640(1)
28. Principal Curves and Umbilic Points
641(20)
28.1 The Differential Equation for the Principal Curves of a Surface
642(3)
28.2 Umbilic Points
645(4)
28.3 The Peterson-Mainardi-Codazzi Equations
649(3)
28.4 Hilbert's Lemma and Liebmann's Theorem
652(2)
28.5 The Fundamental Theorem of Surfaces
654(5)
28.6 Exercises
659(2)
29. Triply Orthogonal Systems of Surfaces
661(20)
29.1 Examples of Triply Orthogonal Systems
662(3)
29.2 Curvilinear Patches and Dupin's Theorem
665(4)
29.3 Elliptic Coordinates
669(7)
29.4 Parabolic Coordinates
676(2)
29.5 Exercises
678(3)
30. Minimal Surfaces
681(20)
30.1 Normal Variation
681(3)
30.2 Examples of Minimal Surfaces
684(10)
30.3 The Gauss Map of a Minimal Surface
694(2)
30.4 Exercises
696(3)
30.5 Animations of Minimal Surfaces
699(2)
31. Minimal Surfaces and Complex Variables
701(34)
31.1 Isothermal Coordinates
702(3)
31.2 Isometric Deformations of Minimal Surfaces
705(5)
31.3 Complex Derivatives
710(4)
31.4 Elementary Complex Vector Algebra
714(2)
31.5 Minimal Curves
716(5)
31.6 Finding Conjugate Minimal Surfaces
721(7)
31.7 Enneper's Surface of Degree n
728(4)
31.8 Exercises
732(2)
31.9 Animations of Minimal Surfaces
734(1)
32. Minimal Surfaces via the Weierstrass Representation
735(26)
32.1 The Weierstrass Representation
736(4)
32.2 Weierstrass Patches via Mathematica
740(1)
32.3 Examples of Weierstrass Patches
741(3)
32.4 Minimal Surfaces with One Planar End
744(3)
32.5 Costa's Minimal Surface
747(13)
32.6 Exercises
760(1)
33. Minimal Surfaces via Bjorling's Formula
761(12)
33.1 Bjorling's Formula
761(3)
33.2 Minimal Surfaces from Plane Curves
764(2)
33.3 Examples of Minimal Surfaces Constructed from Plane Curves
766(6)
33.4 Exercises
772(1)
34. Construction of Surfaces
773(16)
34.1 Parallel Surfaces
773(4)
34.2 Parallel Surfaces to a Mobius Strip
777(2)
34.3 The Shape Operator of a Parallel Surface
779(3)
34.4 A General Construction of a Triply Orthogonal System
782(2)
34.5 Pedal Surfaces
784(2)
34.6 Twisted Surfaces
786(2)
34.7 Exercises
788(1)
35. Canal Surfaces and Cyclides of Dupin
789(30)
35.1 Surfaces Whose Focal Sets are 2-Dimensional
791(7)
35.2 Canal Surfaces
798(11)
35.3 Cyclides of Dupin via Focal Sets
809(8)
35.4 Exercises
817(2)
36. Inversions of Curves and Surfaces
819(18)
36.1 The Definition of Inversion
819(3)
36.2 Inversion of Curves
822(3)
36.3 Inversion of Surfaces
825(5)
36.4 Cyclides of Dupin via Inversions
830(2)
36.5 Liouville's Theorem
832(2)
36.6 Exercises
834(3)
Appendices 837(168)
A General Programs 837(54)
B Curves 891(44)
Parametrically Defined Plane Curves 891(25)
Implicitly Defined Plane Curves 916(7)
Polar Defined Plane Curves 923(4)
Parametrically Defined Space Curves 927(8)
C Surfaces 935(52)
Parametrically Defined Surfaces 935(34)
Implicitly Defined Surfaces 969(5)
Drum Plots 974(2)
Minimal Curves 976(8)
Surface Metrics 984(3)
D Plotting Programs 987(18)
Miscellaneous Plotting Programs 987(7)
Mathematica to Acrospin Programs 994(5)
Mathematica to Geomview Programs 999(6)
Bibliography 1005(16)
Index 1021(20)
Name Index 1041(4)
Miniprogram and Mathematica Command Index 1045


Gray; Alfred University of Maryland, College Park, USA,