Tensor analysis: theory and applications to geometry and mechanics of continua
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Wiley
1964
|
| Edition: | 2. ed. |
| Series: | Applied mathematics series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 361 S. |
| ISBN: | 0471810525 |
Staff View
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| 245 | 1 | 0 | |a Tensor analysis |b theory and applications to geometry and mechanics of continua |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Wiley |c 1964 | |
| 300 | |a XII, 361 S. | ||
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Record in the Search Index
| _version_ | 1804135924839219200 |
|---|---|
| adam_text | CONTENTS
1 LINEAR VECTOR SPACES. MATRICES
1. Coordinate Systems 1
2. The Geometric Concept of a Vector 3
3. Linear Vector Spaces. Dimensionality of Space 6
4. N Dimensional Spaces 8
5. Linear Vector Spaces of n Dimensions 10
6. Complex Linear Vector Spaces 14
7. Summation Convention. Review of Determinants 16
8. Linear Transformations and Matrices 19
9. Linear Transformations in Euclidean 3 space 25
10. Orthogonal Transformation in E:i 27
11. Linear Transformations in /( Dimensional Euclidean Spaces 28
12. Reduction of Matrices to the Diagonal Form 30
13. Real Symmetric Matrices and Quadratic Forms 34
14. Illustrations of Reduction of Quadratic Forms 40
15. Classification and Properties of Quadratic Forms 43
16. Simultaneous Reduction of Two Quadratic Forms to a Sum of Squares 45
17. Unitary Transformations and Hermitean Matrices 47
2 TENSOR THEORY
18. Scope of Tensor Analysis. Invariance 50
19. Transformation of Coordinates 51
20. Properties of Admissible Transformations of Coordinates 53
21. Transformation by Invariance 54
22. Transformation by Covariance and Contravariance 56
23. The Tensor Concept. Contravariant and Covariant Tensors 58
24. Tensor Character of Covariant and Contravariant Laws 62
25. Algebra of Tensors 64
26. Quotient Laws 66
27. Symmetric and Skew Symmetric Tensors 69
28. Relative Tensors 69
29. The Metric Tensor 72
30. The Fundamental and Associated Tensors 74
31. Christoffel s Symbols 75
32. Transformation of Christoffel s Symbols 79
33. Covariant Differentiation of Tensors 81
34. Formulas for Covariant Differentiation 84
35. Ricci s Theorem 86
36. Riemann Christoffel Tensor 86
ix
x CONTENTS
37. Properties of Riemann Christoffel Tensors 89
38. Ricci Tensor. Bianchi Identities. Einstein Tensor 91
39. Riemannian and Euclidean Spaces. Existence Theorem 92
40. The e Systems and the Generalized Kronecker Deltas 97
41. Application of the e Systems to Determinants. Tensor Character of
Generalized Kronecker Deltas 101
3 GEOMETRY
42. Non Euclidean Geometries 105
43. Length of Arc 106
44. Curvilinear Coordinates in £3 112
45. Reciprocal Base Systems. Covariant and Contravariant Vectors 119
46. On the Meaning of Covariant Derivatives 123
47. Intrinsic Differentiation 126
48. Parallel Vector Fields 128
49. Geometry of Space Curves 130
50. Serret Frenet Formulas 134
51. Equations of a Straight Line 137
52. Curvilinear Coordinates on a Surface 138
53. Intrinsic Geometry. First Fundamental Quadratic Form. Metric
Tensor 140
54. Angle between Two Intersecting Curves in a Surface. Element of
Surface Area 144
55. Fundamental Concepts of Calculus of Variations 147
56. Euler s Equation in the Simplest Case 149
57. Euler s Equations for a Functional of Several Arguments 152
58. Geodesies in Rn. 157
59. Geodesic Coordinates 162
60. Parallel Vector Fields in a Surface 163
61. Isometric Surfaces 165
62. The Riemann Christoffel Tensor and the Gaussian Curvature 166
63. The Geodesic Curvature of Surface Curves 169
64. Surfaces in Space 171
65. The Normal Line to the Surface 175
66. Tensor Derivatives 177
67. The Second Fundamental Form of a Surface 180
68. The Integrability Conditions 183
69. Formulas of Weingarten and Equations of Gauss and Codazzi 184
70. The Mean and Total Curvatures of a Surface 186
71. Curves on a Surface. Theorem of Meusnier 187
72. The Principal Curvatures of a Surface 190
73. Parallel Surfaces 195
74. The Gauss Bonnet Theorem 198
75. The n Dimensional Manifolds 202
4 ANALYTICAL MECHANICS
76. Basic Concepts. Kinematics 206
77. Newtonian Laws. Dynamics 207
CONTENTS xi
78. Equations of Motion of a Particle. Work. Energy 209
79. Lagrangean Equations of Motion 212
80. Applications of Lagrangean Equations 215
81. The Symbol of Variation 224
82. Hamilton s Principle 226
83. Integral of Energy 228
84. Principle of Least Action 229
85. Systems of Particles. Generalized Coordinates 233
86. Lagrangean Equations in Generalized Coordinates 235
87. Virtual Work and Generalized Forces 240
88. Nonholonomic Systems 242
89. Illustrative Examples 248
90. Hamilton s Canonical Equations 256
91. Newtonian Law of Gravitation 259
92. Integral Transformation Theorems 263
93. Theorem of Gauss. Solution of Poisson s Equation 268
94. Green s Third Identity. Harmonic Functions 271
95. Functions of Green and Neumann 275
96. Green s Functions for Semi infinite Space and Spherical Regions 278
97. The Problem of Two Bodies 281
5 RELATIVISTIC MECHANICS
98. Invariance of Physical Laws 287
99. Restricted, or Special Theory of Relativity 288
100. Proper or Local Coordinates 292
101. Einstein s Energy Equation 295
102. Restricted Theory. Retrospect and Prospect 297
103. Einstein s Gravitational Equations 298
104. Spherically Symmetric Static Field 300
105. Planetary Orbits 304
106. The Advance of Perihelion 308
107. Concluding Remarks 311
6 MECHANICS OF CONTINUOUS MEDIA
108. Introductory Remarks 313
109. Deformation of a Continuous Medium 314
110. Geometric Interpretation of Strain Tensors £0 and £ 317
111. Strain Quadric. Principal Strains 319
112. Distortion of Volume Elements 322
113. Displacements in Continuous Media 324
114. Equations of Compatibility 326
115. Analysis of Stressed State 326
116. Differential Equations of Equilibrium 330
117. Virtual Work 332
118. Laws of Thermodynamics 336
119. Elastic Media 338
120. Stress Strain Relations in Isotropic Elastic Media 341
121. Equations of Elasticity 343
xii CONTENTS
122. Fluid Mechanics. Equations of Continuity 344
123. Ideal Fluids. Euler s Equations 346
124. Viscous Fluids. Navier s Equations 349
125. Remarks on Turbulent Flows and Dissipative Media 352
Bibliography 353
Index 355
|
| adam_txt |
CONTENTS
1 LINEAR VECTOR SPACES. MATRICES
1. Coordinate Systems 1
2. The Geometric Concept of a Vector 3
3. Linear Vector Spaces. Dimensionality of Space 6
4. N Dimensional Spaces 8
5. Linear Vector Spaces of n Dimensions 10
6. Complex Linear Vector Spaces 14
7. Summation Convention. Review of Determinants 16
8. Linear Transformations and Matrices 19
9. Linear Transformations in Euclidean 3 space 25
10. Orthogonal Transformation in E:i 27
11. Linear Transformations in /( Dimensional Euclidean Spaces 28
12. Reduction of Matrices to the Diagonal Form 30
13. Real Symmetric Matrices and Quadratic Forms 34
14. Illustrations of Reduction of Quadratic Forms 40
15. Classification and Properties of Quadratic Forms 43
16. Simultaneous Reduction of Two Quadratic Forms to a Sum of Squares 45
17. Unitary Transformations and Hermitean Matrices 47
2 TENSOR THEORY
18. Scope of Tensor Analysis. Invariance 50
19. Transformation of Coordinates 51
20. Properties of Admissible Transformations of Coordinates 53
21. Transformation by Invariance 54
22. Transformation by Covariance and Contravariance 56
23. The Tensor Concept. Contravariant and Covariant Tensors 58
24. Tensor Character of Covariant and Contravariant Laws 62
25. Algebra of Tensors 64
26. Quotient Laws 66
27. Symmetric and Skew Symmetric Tensors 69
28. Relative Tensors 69
29. The Metric Tensor 72
30. The Fundamental and Associated Tensors 74
31. Christoffel's Symbols 75
32. Transformation of Christoffel's Symbols 79
33. Covariant Differentiation of Tensors 81
34. Formulas for Covariant Differentiation 84
35. Ricci's Theorem 86
36. Riemann Christoffel Tensor 86
ix
x CONTENTS
37. Properties of Riemann Christoffel Tensors 89
38. Ricci Tensor. Bianchi Identities. Einstein Tensor 91
39. Riemannian and Euclidean Spaces. Existence Theorem 92
40. The e Systems and the Generalized Kronecker Deltas 97
41. Application of the e Systems to Determinants. Tensor Character of
Generalized Kronecker Deltas 101
3 GEOMETRY
42. Non Euclidean Geometries 105
43. Length of Arc 106
44. Curvilinear Coordinates in £3 112
45. Reciprocal Base Systems. Covariant and Contravariant Vectors 119
46. On the Meaning of Covariant Derivatives 123
47. Intrinsic Differentiation 126
48. Parallel Vector Fields 128
49. Geometry of Space Curves ' 130
50. Serret Frenet Formulas 134
51. Equations of a Straight Line 137
52. Curvilinear Coordinates on a Surface 138
53. Intrinsic Geometry. First Fundamental Quadratic Form. Metric
Tensor 140
54. Angle between Two Intersecting Curves in a Surface. Element of
Surface Area 144
55. Fundamental Concepts of Calculus of Variations 147
56. Euler's Equation in the Simplest Case 149
57. Euler's Equations for a Functional of Several Arguments 152
58. Geodesies in Rn. 157
59. Geodesic Coordinates 162
60. Parallel Vector Fields in a Surface 163
61. Isometric Surfaces 165
62. The Riemann Christoffel Tensor and the Gaussian Curvature 166
63. The Geodesic Curvature of Surface Curves 169
64. Surfaces in Space 171
65. The Normal Line to the Surface 175
66. Tensor Derivatives 177
67. The Second Fundamental Form of a Surface 180
68. The Integrability Conditions 183
69. Formulas of Weingarten and Equations of Gauss and Codazzi 184
70. The Mean and Total Curvatures of a Surface 186
71. Curves on a Surface. Theorem of Meusnier 187
72. The Principal Curvatures of a Surface 190
73. Parallel Surfaces 195
74. The Gauss Bonnet Theorem 198
75. The n Dimensional Manifolds 202
4 ANALYTICAL MECHANICS
76. Basic Concepts. Kinematics 206
77. Newtonian Laws. Dynamics 207
CONTENTS xi
78. Equations of Motion of a Particle. Work. Energy 209
79. Lagrangean Equations of Motion 212
80. Applications of Lagrangean Equations 215
81. The Symbol of Variation 224
82. Hamilton's Principle 226
83. Integral of Energy 228
84. Principle of Least Action 229
85. Systems of Particles. Generalized Coordinates 233
86. Lagrangean Equations in Generalized Coordinates 235
87. Virtual Work and Generalized Forces 240
88. Nonholonomic Systems 242
89. Illustrative Examples 248
90. Hamilton's Canonical Equations 256
91. Newtonian Law of Gravitation 259
92. Integral Transformation Theorems 263
93. Theorem of Gauss. Solution of Poisson's Equation 268
94. Green's Third Identity. Harmonic Functions 271
95. Functions of Green and Neumann 275
96. Green's Functions for Semi infinite Space and Spherical Regions 278
97. The Problem of Two Bodies 281
5 RELATIVISTIC MECHANICS
98. Invariance of Physical Laws 287
99. Restricted, or Special Theory of Relativity 288
100. Proper or Local Coordinates 292
101. Einstein's Energy Equation 295
102. Restricted Theory. Retrospect and Prospect 297
103. Einstein's Gravitational Equations 298
104. Spherically Symmetric Static Field 300
105. Planetary Orbits 304
106. The Advance of Perihelion 308
107. Concluding Remarks 311
6 MECHANICS OF CONTINUOUS MEDIA
108. Introductory Remarks 313
109. Deformation of a Continuous Medium 314
110. Geometric Interpretation of Strain Tensors £0 and £ 317
111. Strain Quadric. Principal Strains 319
112. Distortion of Volume Elements 322
113. Displacements in Continuous Media 324
114. Equations of Compatibility 326
115. Analysis of Stressed State 326
116. Differential Equations of Equilibrium 330
117. Virtual Work 332
118. Laws of Thermodynamics 336
119. Elastic Media 338
120. Stress Strain Relations in Isotropic Elastic Media 341
121. Equations of Elasticity 343
xii CONTENTS
122. Fluid Mechanics. Equations of Continuity 344
123. Ideal Fluids. Euler's Equations 346
124. Viscous Fluids. Navier's Equations 349
125. Remarks on Turbulent Flows and Dissipative Media 352
Bibliography 353
Index 355 |
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| id | DE-604.BV021957841 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:08:14Z |
| indexdate | 2024-07-09T20:48:14Z |
| institution | BVB |
| isbn | 0471810525 |
| language | English |
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| physical | XII, 361 S. |
| publishDate | 1964 |
| publishDateSearch | 1964 |
| publishDateSort | 1964 |
| publisher | Wiley |
| record_format | marc |
| series2 | Applied mathematics series |
| spelling | Sokolnikoff, Ivan S. Verfasser aut Tensor analysis theory and applications to geometry and mechanics of continua 2. ed. New York [u.a.] Wiley 1964 XII, 361 S. txt rdacontent n rdamedia nc rdacarrier Applied mathematics series Algebra larpcal Analyse (wiskunde) gtt Mathematische fysica gtt Tensoren gtt Toepassingen gtt Calculus of tensors Anwendung (DE-588)4196864-5 gnd rswk-swf Tensoranalysis (DE-588)4204323-2 gnd rswk-swf Tensoranalysis (DE-588)4204323-2 s Anwendung (DE-588)4196864-5 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015172991&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Sokolnikoff, Ivan S. Tensor analysis theory and applications to geometry and mechanics of continua Algebra larpcal Analyse (wiskunde) gtt Mathematische fysica gtt Tensoren gtt Toepassingen gtt Calculus of tensors Anwendung (DE-588)4196864-5 gnd Tensoranalysis (DE-588)4204323-2 gnd |
| subject_GND | (DE-588)4196864-5 (DE-588)4204323-2 |
| title | Tensor analysis theory and applications to geometry and mechanics of continua |
| title_auth | Tensor analysis theory and applications to geometry and mechanics of continua |
| title_exact_search | Tensor analysis theory and applications to geometry and mechanics of continua |
| title_exact_search_txtP | Tensor analysis theory and applications to geometry and mechanics of continua |
| title_full | Tensor analysis theory and applications to geometry and mechanics of continua |
| title_fullStr | Tensor analysis theory and applications to geometry and mechanics of continua |
| title_full_unstemmed | Tensor analysis theory and applications to geometry and mechanics of continua |
| title_short | Tensor analysis |
| title_sort | tensor analysis theory and applications to geometry and mechanics of continua |
| title_sub | theory and applications to geometry and mechanics of continua |
| topic | Algebra larpcal Analyse (wiskunde) gtt Mathematische fysica gtt Tensoren gtt Toepassingen gtt Calculus of tensors Anwendung (DE-588)4196864-5 gnd Tensoranalysis (DE-588)4204323-2 gnd |
| topic_facet | Algebra Analyse (wiskunde) Mathematische fysica Tensoren Toepassingen Calculus of tensors Anwendung Tensoranalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015172991&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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