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E-raamat: Visual Introduction to Differential Forms and Calculus on Manifolds

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 03-Nov-2018
  • Kirjastus: Birkhauser Verlag AG
  • Keel: eng
  • ISBN-13: 9783319969923
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Visual Introduction to Differential Forms and Calculus on ManifoldsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 03-Nov-2018
  • Kirjastus: Birkhauser Verlag AG
  • Keel: eng
  • ISBN-13: 9783319969923
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This book explains and helps readers to develop geometric intuition as it relates to differential forms. It includes over 250 figures to aid understanding and enable readers to visualize the concepts being discussed. The author gradually builds up to the basic ideas and concepts so that definitions, when made, do not appear out of nowhere, and both the importance and role that theorems play is evident as or before they are presented. With a clear writing style and easy-to- understand motivations for each topic, this book is primarily aimed at second- or third-year undergraduate math and physics students with a basic knowledge of vector calculus and linear algebra.

Arvustused

 The reviewer recommends young mathematics and physics majors to open the book and to keep it on their bookshelves. Indeed, the reviewer even envies young students who can study differential forms with such a fascinating book. (Hirokazu Nishimura, zbMath 1419.58001, 2019)

1 Background Material
1(30)
1.1 Review of Vector Spaces
1(15)
1.2 Volume and Determinants
16(7)
1.3 Derivatives of Multivariable Functions
23(4)
1.4 Summary, References, and Problems
27(4)
1.4.1 Summary
27(1)
1.4.2 References and Further Reading
28(1)
1.4.3 Problems
28(3)
2 An Introduction to Differential Forms
31(38)
2.1 Coordinate Functions
31(6)
2.2 Tangent Spaces and Vector Fields
37(6)
2.3 Directional Derivatives
43(10)
2.4 Differential One-Forms
53(12)
2.5 Summary, References, and Problems
65(4)
2.5.1 Summary
65(1)
2.5.2 References and Further Reading
66(1)
2.5.3 Problems
66(3)
3 The Wedgeproduct
69(38)
3.1 Area and Volume with the Wedgeproduct
69(13)
3.2 General Two-Forms and Three-Forms
82(6)
3.3 The Wedgeproduct of n-Forms
88(9)
3.3.1 Algebraic Properties
88(2)
3.3.2 Simplifying Notation
90(3)
3.3.3 The General Formula
93(4)
3.4 The Interior Product
97(3)
3.5 Summary, References, and Problems
100(7)
3.5.1 Summary
100(2)
3.5.2 References and Further Reading
102(1)
3.5.3 Problems
102(5)
4 Exterior Differentiation
107(44)
4.1 An Overview of the Exterior Derivative
107(2)
4.2 The Local Formula
109(3)
4.3 The Axioms of Exterior Differentiation
112(2)
4.4 The Global Formula
114(16)
4.4.1 Exterior Differentiation with Constant Vector Fields
114(7)
4.4.2 Exterior Differentiation with Non-Constant Vector Fields
121(9)
4.5 Another Geometric Viewpoint
130(12)
4.6 Exterior Differentiation Examples
142(5)
4.7 Summary, References, and Problems
147(4)
4.7.1 Summary
147(1)
4.7.2 References and Further Reading
148(1)
4.7.3 Problems
149(2)
5 Visualizing One-, Two-, and Three-Forms
151(38)
5.1 One- and Two-Forms in R2
151(9)
5.2 One-Forms in R3
160(6)
5.3 Two-Forms in R3
166(9)
5.4 Three-Forms in R3
175(1)
5.5 Pictures of Forms on Manifolds
175(4)
5.6 A Visual Introduction to the Hodge Star Operator
179(7)
5.7 Summary, References, and Problems
186(3)
5.7.1 Summary
186(1)
5.7.2 References and Further Reading
187(1)
5.7.3 Problems
187(2)
6 Push-Forwards and Pull-Backs
189(40)
6.1 Coordinate Change: A Linear Example
189(7)
6.2 Push-Forwards of Vectors
196(5)
6.3 Pull-Backs of Volume Forms
201(5)
6.4 Polar Coordinates
206(7)
6.5 Cylindrical and Spherical Coordinates
213(4)
6.6 Pull-Backs of Differential Forms
217(6)
6.7 Some Useful Identities
223(3)
6.8 Summary, References, and Problems
226(3)
6.8.1 Summary
226(1)
6.8.2 References and Further Reading
227(1)
6.8.3 Problems
227(2)
7 Changes of Variables and Integration of Forms
229(30)
7.1 Integration of Differential Forms
229(6)
7.2 A Simple Example
235(5)
7.3 Polar, Cylindrical, and Spherical Coordinates
240(5)
7.3.1 Polar Coordinates Example
240(3)
7.3.2 Cylindrical Coordinates Example
243(1)
7.3.3 Spherical Coordinates Example
244(1)
7.4 Integration of Differential Forms on Parameterized Surfaces
245(9)
7.4.1 Line Integrals
246(5)
7.4.2 Surface Integrals
251(3)
7.5 Summary, References, and Problems
254(5)
7.5.1 Summary
254(1)
7.5.2 References and Further Reading
255(1)
7.5.3 Problems
255(4)
8 Poincare Lemma
259(18)
8.1 Introduction to the Poincare Lemma
259(2)
8.2 The Base Case and a Simple Example Case
261(7)
8.3 The General Case
268(7)
8.4 Summary, References, and Problems
275(2)
8.4.1 Summary
275(1)
8.4.2 References and Further Reading
275(1)
8.4.3 Problems
275(2)
9 Vector Calculus and Differential Forms
277(32)
9.1 Divergence
277(7)
9.2 Curl
284(9)
9.3 Gradient
293(1)
9.4 Upper and Lower Indices, Sharps, and Flats
294(4)
9.5 Relationship to Differential Forms
298(7)
9.5.1 Grad, Curl, Div and Exterior Differentiation
298(4)
9.5.2 Fundamental Theorem of Line Integrals
302(1)
9.5.3 Vector Calculus Stokes' Theorem
303(1)
9.5.4 Divergence Theorem
304(1)
9.6 Summary, References, and Problems
305(4)
9.6.1 Summary
305(1)
9.6.2 References and Further Reading
306(1)
9.6.3 Problems
307(2)
10 Manifolds and Forms on Manifolds
309(28)
10.1 Definition of a Manifold
309(4)
10.2 Tangent Space of a Manifold
313(10)
10.3 Push-Forwards and Pull-Backs on Manifolds
323(3)
10.4 Calculus on Manifolds
326(6)
10.4.1 Differentiation on Manifolds
327(1)
10.4.2 Integration on Manifolds
328(4)
10.5 Summary, References, and Problems
332(5)
10.5.1 Summary
332(2)
10.5.2 References and Further Reading
334(1)
10.5.3 Problems
334(3)
11 Generalized Stokes' Theorem
337(32)
11.1 The Unit Cube Ik
337(16)
11.2 The Base Case: Stokes' Theorem on Ik
353(5)
11.3 Manifolds Parameterized by Ik
358(1)
11.4 Stokes' Theorem on Chains
359(3)
11.5 Extending the Parameterizations
362(1)
11.6 Visualizing Stokes' Theorem
363(3)
11.7 Summary, References, and Problems
366(3)
11.7.1 Summary
366(1)
11.7.2 References and Further Reading
366(1)
11.7.3 Problems
366(3)
12 An Example: Electromagnetism
369(26)
12.1 Gauss's Laws for Electric and Magnetic Fields
369(6)
12.2 Faraday's Law and the Ampere-Maxwell Law
375(5)
12.3 Special Relativity and Hodge Duals
380(4)
12.4 Differential Forms Formulation
384(6)
12.5 Summary, References, and Problems
390(5)
12.5.1 Summary
390(2)
12.5.2 References and Further Reading
392(1)
12.5.3 Problems
392(3)
A Introduction to Tensors
395(40)
A.1 An Overview of Tensors
395(1)
A.2 Rank One Tensors
396(8)
A.3 Rank-Two Tensors
404(3)
A.4 General Tensors
407(2)
A.5 Differential Forms as Skew-Symmetric Tensors
409(2)
A.6 The Metric Tensor
411(3)
A.7 Lie Derivatives of Tensor Fields
414(17)
A.8 Summary and References
431(4)
A.8.1 Summary
431(3)
A.8.2 References and Further Reading
434(1)
B Some Applications of Differential Forms
435(28)
B.1 Introduction to de Rham Cohomology
435(4)
B.2 De Rham Cohomology: A Few Simple Examples
439(4)
B.3 Symplectic Manifolds and the Connonical Symplectic Form
443(6)
B.4 The Darboux Theorem
449(4)
B.5 A Taste of Geometric Mechanics
453(6)
B.6 Summary and References
459(4)
B.6.1 Summary
459(2)
B.6.2 References and Further Reading
461(2)
References 463(2)
Index 465
Jon Pierre Fortney, Zayed University, Dubai, United Arab Emirates.  
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Püsilink: https://www.kriso.ee/db/97833199699232e.html