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E-raamat: Introduction to Differential Geometry with Tensor Applications

Edited by (University of Calcutta, India; Tripura University, India)
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INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH TENSOR APPLICATIONS

This is the only volume of its kind to explain, in precise and easy-to-understand language, the fundamentals of tensors and their applications in differential geometry and analytical mechanics with examples for practical applications and questions for use in a course setting.

Introduction to Differential Geometry with Tensor Applications discusses the theory of tensors, curves and surfaces and their applications in Newtonian mechanics. Since tensor analysis deals with entities and properties that are independent of the choice of reference frames, it forms an ideal tool for the study of differential geometry and also of classical and celestial mechanics. This book provides a profound introduction to the basic theory of differential geometry: curves and surfaces and analytical mechanics with tensor applications. The author has tried to keep the treatment of the advanced material as lucid and comprehensive as possible, mainly by including utmost detailed calculations, numerous illustrative examples, and a wealth of complementing exercises with complete solutions making the book easily accessible even to beginners in the field.

Groundbreaking and thought-provoking, this volume is an outstanding primer for modern differential geometry and is a basic source for a profound introductory course or as a valuable reference. It can even be used for self-study, by students or by practicing engineers interested in the subject.

Whether for the student or the veteran engineer or scientist, Introduction to Differential Geometry with Tensor Applications is a must-have for any library.

This outstanding new volume:

  • Presents a unique perspective on the theories in the field not available anywhere else
  • Explains the basic concepts of tensors and matrices and their applications in differential geometry and analytical mechanics
  • Is filled with hundreds of examples and unworked problems, useful not just for the student, but also for the engineer in the field
  • Is a valuable reference for the professional engineer or a textbook for the engineering student
Preface xv
About the Book xvii
Introduction 1(6)
Part I: Tensor Theory 7(186)
1 Preliminaries
9(16)
1.1 Introduction
9(1)
1.2 Systems of Different Orders
9(1)
1.3 Summation Convention Certain Index
10(1)
1.3.1 Dummy Index
11(1)
1.3.2 Free Index
11(1)
1.4 Kronecker Symbols
11(3)
1.5 Linear Equations
14(1)
1.6 Results on Matrices and Determinants of Systems
15(3)
1.7 Differentiation of a Determinant
18(1)
1.8 Examples
19(4)
1.9 Exercises
23(2)
2 Tensor Algebra
25(48)
2.1 Introduction
25(1)
2.2 Scope of Tensor Analysis
25(2)
2.2.1 n-Dimensional Space
26(1)
2.3 Transformation of Coordinates in Sn
27(4)
2.3.1 Properties of Admissible Transformation of Coordinates
30(1)
2.4 Transformation by Invariance
31(1)
2.5 Transformation by Covariant Tensor and Contravariant Tensor
32(2)
2.6 The Tensor Concept: Contravariant and Covariant Tensors
34(9)
2.6.1 Covariant Tensors
34(1)
2.6.2 Contravariant Vectors
35(5)
2.6.3 Tensor of Higher Order
40(3)
2.6.3.1 Contravariant Tensors of Order Two
40(1)
2.6.3.2 Covariant Tensor of Order Two
41(1)
2.6.3.3 Mixed Tensors of Order Two
42(1)
2.7 Algebra of Tensors
43(2)
2.7.1 Equality of Two Tensors of Same Type
45(1)
2.8 Symmetric and Skew-Symmetric Tensors
45(6)
2.8.1 Symmetric Tensors
45(1)
2.8.2 Skew-Symmetric Tensors
46(5)
2.9 Outer Multiplication and Contraction
51(5)
2.9.1 Outer Multiplication
51(2)
2.9.2 Contraction of a Tensor
53(1)
2.9.3 Inner Product of Two Tensors
54(2)
2.10 Quotient Law of Tensors
56(2)
2.11 Reciprocal Tensor of a Tensor
58(2)
2.12 Relative Tensor, Cartesian Tensor, Affine Tensor, and Isotropic Tensors
60(5)
2.12.1 Relative Tensors
60(3)
2.12.2 Cartesian Tensors
63(1)
2.12.3 Affine Tensors
63(1)
2.12.4 Isotropic Tensor
64(1)
2.12.5 Pseudo-Tensors
64(1)
2.13 Examples
65(6)
2.14 Exercises
71(2)
3 Riemannian Metric
73(24)
3.1 Introduction
73(1)
3.2 The Metric Tensor
74(1)
3.3 Conjugate Tensor
75(2)
3.4 Associated Tensors
77(7)
3.5 Length of a Vector
84(2)
3.5.1 Length of Vector
84(1)
3.5.2 Unit Vector
85(1)
3.5.3 Null Vector
86(1)
3.6 Angle Between Two Vectors
86(2)
3.6.1 Orthogonality of Two Vectors
87(1)
3.7 Hypersurface
88(1)
3.8 Angle Between Two Coordinate Hypersurfaces
89(6)
3.9 Exercises
95(2)
4 Tensor Calculus
97(46)
4.1 Introduction
97(1)
4.2 Christoffel Symbols
97(13)
4.2.1 Properties of Christoffel Symbols
98(12)
4.3 Transformation of Christoffel Symbols
110(3)
4.3.1 Law of Transformation of Christoffel Symbols of 1st Kind
110(1)
4.3.2 Law of Transformation of Christoffel Symbols of 2nd Kind
111(2)
4.4 Covariant Differentiation of Tensor
113(16)
4.4.1 Covariant Derivative of Covariant Tensor
114(1)
4.4.2 Covariant Derivative of Contravariant Tensor
115(1)
4.4.3 Covariant Derivative of Tensors of Type (0,2)
116(2)
4.4.4 Covariant Derivative of Tensors of Type (2,0)
118(2)
4.4.5 Covariant Derivative of Mixed Tensor of Type (s, r)
120(1)
4.4.6 Covariant Derivatives of Fundamental Tensors and the Kronecker Delta
120(2)
4.4.7 Formulas for Covariant Differentiation
122(1)
4.4.8 Covariant Differentiation of Relative Tensors
123(6)
4.5 Gradient, Divergence, and Curl
129(12)
4.5.1 Gradient
130(1)
4.5.2 Divergence
130(6)
4.5.2.1 Divergence of a Mixed Tensor (1,1)
132(4)
4.5.3 Laplacian of an Invariant
136(1)
4.5.4 Curl of a Covariant Vector
137(4)
4.6 Exercises
141(2)
5 Riemannian Geometry
143(34)
5.1 Introduction
143(1)
5.2 Riemannian-Christoffel Tensor
143(7)
5.3 Properties of Riemann-Christoffel Tensors
150(9)
5.3.1 Space of Constant Curvature
158(1)
5.4 Ricci Tensor, Bianchi Identities, Einstein Tensors
159(11)
5.4.1 Ricci Tensor
159(1)
5.4.2 Bianchi Identity
160(6)
5.4.3 Einstein Tensor
166(4)
5.5 Einstein Space
170(1)
5.6 Riemannian and Euclidean Spaces
171(4)
5.6.1 Riemannian Spaces
171(3)
5.6.2 Euclidean Spaces
174(1)
5.7 Exercises
175(2)
6 The e-Systems and the Generalized Kronecker Deltas
177(16)
6.1 Introduction
177(1)
6.2 e-Systems
177(4)
6.3 Generalized Kronecker Delta
181(2)
6.4 Contraction of δijkαβγ
183(2)
6.5 Application of e-Systems to Determinants and Tensor Characters of Generalized Kronecker Deltas
185(7)
6.5.1 Curl of Covariant Vector
189(1)
6.5.2 Vector Product of Two Covariant Vectors
190(2)
6.6 Exercises
192(1)
Part II: Differential Geometry 193(204)
7 Curvilinear Coordinates in Space
195(26)
7.1 Introduction
195(1)
7.2 Length of Arc
195(5)
7.3 Curvilinear Coordinates in E3
200(10)
7.3.1 Coordinate Surfaces
201(1)
7.3.2 Coordinate Curves
202(3)
7.3.3 Line Element
205(1)
7.3.4 Length of a Vector
206(1)
7.3.5 Angle Between Two Vectors
207(3)
7.4 Reciprocal Base Systems
210(6)
7.5 Partial Derivative
216(3)
7.6 Exercises
219(2)
8 Curves in Space
221(44)
8.1 Introduction
221(1)
8.2 Intrinsic Differentiation
221(5)
8.3 Parallel Vector Fields
226(2)
8.4 Geometry of Space Curves
228(5)
8.4.1 Plane
231(2)
8.5 Serret-Frenet Formula
233(19)
8.5.1 Bertrand Curves
235(17)
8.6 Equations of a Straight Line
252(2)
8.7 Helix
254(8)
8.7.1 Cylindrical Helix
256(2)
8.7.2 Circular Helix
258(4)
8.8 Exercises
262(3)
9 Intrinsic Geometry of Surfaces
265(56)
9.1 Introduction
265(1)
9.2 Curvilinear Coordinates on a Surface
265(2)
9.3 Intrinsic Geometry: First Fundamental Quadratic Form
267(5)
9.3.1 Contravariant Metric Tensor
270(2)
9.4 Angle Between Two Intersecting Curves on a Surface
272(5)
9.4.1 Pictorial Interpretation
274(3)
9.5 Geodesic in Rn
277(12)
9.6 Geodesic Coordinates
289(2)
9.7 Parallel Vectors on a Surface
291(1)
9.8 Isometric Surface
292(2)
9.8.1 Developable
293(1)
9.9 The Riemannian-Christoffel Tensor and Gaussian Curvature
294(14)
9.9.1 Einstein Curvature
296(12)
9.10 The Geodesic Curvature
308(11)
9.11 Exercises
319(2)
10 Surfaces in Space
321(28)
10.1 Introduction
321(1)
10.2 The Tangent Vector
321(3)
10.3 The Normal Line to the Surface
324(5)
10.4 Tensor Derivatives
329(3)
10.5 Second Fundamental Form of a Surface
332(2)
10.5.1 Equivalence of Definition of Tensor bαβ
333(1)
10.6 The Integrability Condition
334(3)
10.7 Formulas of Weingarten
337(2)
10.7.1 Third Fundamental Form
338(1)
10.8 Equations of Gauss and Codazzi
339(2)
10.9 Mean and Total Curvatures of a Surface
341(6)
10.10 Exercises
347(2)
11 Curves on a Surface
349(32)
11.1 Introduction
349(1)
11.2 Curve on a Surface: Theorem of Meusnier
350(8)
11.2.1 Theorem of Meusnier
353(5)
11.3 The Principal Curvatures of a Surface
358(18)
11.3.1 Umbillic Point
360(1)
11.3.2 Lines of Curvature
361(1)
11.3.3 Asymptotic Lines
362(14)
11.4 Rodrigue's Formula
376(3)
11.5 Exercises
379(2)
12 Curvature of Surface
381(16)
12.1 Introduction
381(1)
12.2 Surface of Positive and Negative Curvatures
381(2)
12.3 Parallel Surfaces
383(4)
12.3.1 Computation of aαβ and bαβ
383(4)
12.4 The Gauss-Bonnet Theorem
387(4)
12.5 The n-Dimensional Manifolds
391(3)
12.6 Hypersurfaces
394(1)
12.7 Exercises
395(2)
Part III: Analytical Mechanics 397(72)
13 Classical Mechanics
399(48)
13.1 Introduction
399(1)
13.2 Newtonian Laws of Motion
399(2)
13.3 Equations of Motion of Particles
401(2)
13.4 Conservative Force Field
403(2)
13.5 Lagrangean Equations of Motion
405(6)
13.6 Applications of Lagrangean Equations
411(12)
13.7 Himilton's Principle
423(4)
13.8 Principle of Least Action
427(3)
13.9 Generalized Coordinates
430(2)
13.10 Lagrangean Equations in Generalized Coordinates
432(6)
13.11 Divergence Theorem, Green's Theorem, Laplacian Operator, and Stoke's Theorem in Tensor Notation
438(4)
13.12 Hamilton's Canonical Equations
442(2)
13.12.1 Generalized Momenta
443(1)
13.13 Exercises
444(3)
14 Newtonian Law of Gravitations
447(22)
14.1 Introduction
447(1)
14.2 Newtonian Laws of Gravitation
447(4)
14.3 Theorem of Gauss
451(2)
14.4 Poisson's Equation
453(1)
14.5 Solution of Poisson's Equation
454(2)
14.6 The Problem of Two Bodies
456(6)
14.7 The Problem of Three Bodies
462(5)
14.8 Exercises
467(2)
Appendix A: Answers to Even-Numbered Exercises 469(4)
References 473(2)
Index 475
Dipankar De, PhD, received his BSc and MSc in mathematics from the University of Calcutta, India and his PhD in mathematics from Tripura University, India. He has over 40 years of teaching experience and is an associate professor and guest lecturer in India. He has published many research papers in various reputed journal in the field of fuzzy mathematics and differential geometry.
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