Application of tensor analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Dover
1957
|
| Schriftenreihe: | Dover books on intermediate and advanced mathematics.
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Früher u.d.T.: Applications of the absolute differential calculus. - (Früher u.d.T.:) Applications of the absolute differential calculus Umschlagtitel: Applications of tensor analysis |
| Beschreibung: | XII,318 S.m.Tab. |
| ISBN: | 0486603733 |
Internformat
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| 100 | 1 | |a McConnell, Albert J. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Application of tensor analysis |c A. J. MacConnell |
| 264 | 1 | |a New York |b Dover |c 1957 | |
| 300 | |a XII,318 S.m.Tab. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Dover books on intermediate and advanced mathematics. | |
| 500 | |a Früher u.d.T.: Applications of the absolute differential calculus. - (Früher u.d.T.:) Applications of the absolute differential calculus | ||
| 500 | |a Umschlagtitel: Applications of tensor analysis | ||
| 650 | 7 | |a Analyse (wiskunde) |2 gtt | |
| 650 | 4 | |a Calcul tensoriel | |
| 650 | 7 | |a Tensoren |2 gtt | |
| 650 | 7 | |a Toepassingen |2 gtt | |
| 650 | 4 | |a Calculus of tensors | |
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Datensatz im Suchindex
| _version_ | 1813154999157391360 |
|---|---|
| adam_text |
CONTENTS.
PART I. — ALGEBRAIC PRELIMINARIES.
CHAPTER I.
NOTATION AND DEFINITIONS.
1. The indicial notation * 1
2. The summation convention 3
3. Addition, multiplication, and contraction of systems . 5
4. Symmetric and skew symmetric systems 6
5. The skew symmetric three systems and the Kronecker deltas 7
DETERMINANTS.
6. The determinant formed by a double system a\ • • 10
7. The cofactors of the elements in a determinant 12
8. linear equations 14
9. Corresponding formula for the system amn . 15
10. Positive definite quadratic forms. The determinantal equation 16
CHAPTER H.
TENSOR ANALYSIS.
1. Linear transformations .19
2. Invariants, contravariant and covariant vectors . 20
3. Tensors of any order 22
4. Addition, multiplication and contraction of tensors 24
5. The quotient law of tensors 26
6. Relative or weighted tensors 28
7. General functional transformations 30
8. Tensors with respect to the general functional transformation 32
PART II. —ALGEBRAIC GEOMETRY.
CHAPTER IE.
RECTILINEAR COORDINATES.
1. Coordinates and tensors 35
2. Contravariant vectors and displacements 37
3. The unit points and the geometrical interpretation of rectilinear
coordinates .•. . .3g
Vii
viii CONTENTS.
4. The distance between two points and the fundamental double ten¬
sor. The c systems 41
5. The angle between two directions; orthogonality 43
6. Associated tensors 44
7. Scalar and vector products of vectors 46
8. Areas and volumes .49
CHAPTER IV.
THE PLANE.
1. The equation of a plane 52
2. The perpendicular distance from a point to a plane 54
3. The intersection of two planes 56
4. The intersection of three planes 58
5. Plane coordinates 61
6. Systems of planes 63
7. The equation of a point 64
CHAPTER V.
THE STRAIGHT LINE.
1. The point equations of the straight line 68
2. The relations of two straight lines 69
3. The six coordinates of a straight line 71
4. The plane equations of a straight line . 72
CHAPTER VI.
THE QUADRIC CONE AND THE CONIC.
1. The equation of a quadric cone 75
2. The equation of a conic 76
3. The tangent plane to a cone 78
4. Poles and polar planes with respect to a co;u 80
5. The canonical equation of a cone 81
6. The principal axes of a cone . . 83
7. The classification of cones 85
CHAPTER VII.
SYSTEMS OF CONES AND CONICS.
1. The equation of a system of cones with a common vertex . • 88
2. The common polar directions of a family of cones 89
3. The canonical forms of the equation of a family of cones • 92
4. The theory of elementary divisors 97
6. Analytical discrimination of the cases 100
CONTENTS. fa
CHAPTER Tin.
CENTRAL QUADRICS.
1. The point equation of a central quadric • 104
2. The tangential equation of a central quadric 105
3. Canonical form of the equation of a quadric. Principal axes 107
4. Classification of the central quadrics 108
5. Confocal quadrics • 110
CHAPTER IX.
THE GENERAL QUADRIC.
1. The general equation of a quadric • 113
2. The centre 114
3. The reduction of the equation of a quadrio . 115
CHAPTER X.
AFFINE TRANSFORMATIONS.
1. Affine transformations • 120
2. The quadric of a transformation 121
3. Pure strain 123
4. Rigid body displacements 124
5. Infinitesimal deformations 126
PART. III. — DIFFERENTIAL GEOMETRY.
CHAPTER XI.
CURVILINEAR COORDINATES.
1. General coordinate systems 130
2. Tensor fields 133
3. The line element and the metric tensor. The e systems 134
4. The angle between two directions 136
CHAPTER XII.
COVARIANT DIFFERENTIATION.
1. A parallel field of vectors. The Christoffel symbols 140
2. The intrinsic and covariant derivation of vectors . 143
3. The intrinsic and covariant derivatives of tensors . 146
4. Conservation of the rules of the ordinary differential calculus.
Ricci's lemma 148
5. The divergence and curl of a vector. The Laplacian . 151
3. The Riemann Christoffel tensor. The Lame relations • 152
x CONTENTS.
CHAPTER XIII.
CURVES IN SPACE.
1. The tangent vector to a curve 156
2. Normal vectors. The principal normal and binormal . 157
3. The Frenet formulae 159
4. Parallel vectors along a curve. The straight line . 160
CHAPTER XTV.
INTRINSIC GEOMETRY OF A SURFACE.
1. Curvilinear coordinates on a surface 163
2. The conventions regarding Greek indices. Surface tensors 164
3. The element of length and the metric tensor . 167
4. Directions on a surface. Angle between two directions 168
5. The equations of a geodesic 171
6. The transformation of the Christoffel symbols. Geodesic coordin¬
ates 175
7. Parallelism with respect to a surface 178
8. Intrinsic and covariant differentiation of surface tensors 180
9. The Riemann Christoffel tensor. The Gaussian curvature of a sur¬
face 182
10. The geodesic curvature of a curve on a surface 184
11. Beltrami's differential parameters 186
12. Green's theorem on a surface . 188
CHAPTER XV.
THE FUNDAMENTAL FORMULAE OF A SURFACE.
1. Notation 193
2. The tangent vectors to a surface 194
3. The first groundform of a surface 195
4. The normal vector to the surface . 196
5. The tensor derivation of tensors 197
6. Gauss's formulae. The second groundform of a surface 200
7. Weingarten's formulae. The third groundform of a surface 201
8. The equations of Gauss and Codazzi 203
CHAPTER XVI.
CURVES ON A SURFACE.
1. The equations of a curve on a surface 207
2. Meusnier's theorem . 208
3. The principal curvatures. Gauss's theorem 210
4. The lines of curvature 211
5. The asymptotic lines. Enneper's formula 213
6. The geodesic torsion of a curve on a surface • • • 214
CONTENTS. jd
PART IV. — APPLIED MATHEMATICS.
CHAPTER XVII.
DYNAMICS OF A PARTICLE.
1. The equations of motion 218
2. Work and energy. Lagrange's equations of motion . 220
3. Particle on a curve 223
4. Particle on a surface 226
5. The principle of least action. Trajectories as geodesies . 228
CHAPTER XVHI.
DYNAMICS OF RIGID BODIES.
SECTION A — RECTILINEAR COORDINATES.
1. Moments of Inertia 233
2. The equations of motion 235
3. Moving axes. Euler's equations . 238
SECTION B — THE GEOMETRY OF DYNAMICS.
4. Generalised coordinates of a dynamical system . 240
5. The equations of motion in generalised coordinates . . . 242
6. The manifold of configurations 245
7. The kinematical line element 246
8. The dynamical trajectories of the manifold of configurations 247
9. The principle of stationary action. The action line element 249
CHAPTER XIX.
ELECTRICITY AND MAGNETISM.
1. Green's theorem 255
2. Stokes's theorem 258
3. The electrostatic field 259
4. Dielectrics 261
5. The magnetostatic field 263
6. The electromagnetic equations 265
CHAPTER XX.
MECHANICS OF CONTINUOUS MEDIA.
1. Infinitesimal strain . 271
2. Analysis of stress 274
3. Equations of motion for a perfect fluid 276
4. The equations of elasticity 278
5. The motion of a viscous fluid 280
xii CONTENTS.
CHAPTER XXI.
THE SPECIAL THEORY OF RELATIVITY.
1. The four dimensional manifold 285
2. Generalised coordinates in space time 286
3. The principle of special relativity. The interval and the funda¬
mental quadratic form 288
4. Local coordinate systems and their transformations 292
5. Belativistic dynamics of a particle 294
6. Dynamics of a continuous medium 296
7. The electromagnetic equations • 298
APPENDIX.
ORTHOGONAL CURVILINEAR COORDINATES IN
MATHEMATICAL PHYSICS.
1. The classical notation • 303
2. The physical components of vectors and tensors . 304
3. Dynamics 305
4. Electricity 306
5. Elasticity 307
6. Hydrodynamics . 309
BIBLIOGRAPHY 314
INDEX 315 |
| any_adam_object | 1 |
| author | McConnell, Albert J. |
| author_facet | McConnell, Albert J. |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 370 SK 400 |
| ctrlnum | (OCoLC)528595 (DE-599)BVBBV001932059 |
| dewey-full | 517.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 517 - [Unassigned] |
| dewey-raw | 517.2 |
| dewey-search | 517.2 |
| dewey-sort | 3517.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| isbn | 0486603733 |
| language | English |
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| physical | XII,318 S.m.Tab. |
| publishDate | 1957 |
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| publisher | Dover |
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| spelling | McConnell, Albert J. Verfasser aut Application of tensor analysis A. J. MacConnell New York Dover 1957 XII,318 S.m.Tab. txt rdacontent n rdamedia nc rdacarrier Dover books on intermediate and advanced mathematics. Früher u.d.T.: Applications of the absolute differential calculus. - (Früher u.d.T.:) Applications of the absolute differential calculus Umschlagtitel: Applications of tensor analysis Analyse (wiskunde) gtt Calcul tensoriel Tensoren gtt Toepassingen gtt Calculus of tensors Tensoranalysis (DE-588)4204323-2 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Tensoranalysis (DE-588)4204323-2 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001259358&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | McConnell, Albert J. Application of tensor analysis Analyse (wiskunde) gtt Calcul tensoriel Tensoren gtt Toepassingen gtt Calculus of tensors Tensoranalysis (DE-588)4204323-2 gnd |
| subject_GND | (DE-588)4204323-2 (DE-588)4151278-9 |
| title | Application of tensor analysis |
| title_auth | Application of tensor analysis |
| title_exact_search | Application of tensor analysis |
| title_full | Application of tensor analysis A. J. MacConnell |
| title_fullStr | Application of tensor analysis A. J. MacConnell |
| title_full_unstemmed | Application of tensor analysis A. J. MacConnell |
| title_short | Application of tensor analysis |
| title_sort | application of tensor analysis |
| topic | Analyse (wiskunde) gtt Calcul tensoriel Tensoren gtt Toepassingen gtt Calculus of tensors Tensoranalysis (DE-588)4204323-2 gnd |
| topic_facet | Analyse (wiskunde) Calcul tensoriel Tensoren Toepassingen Calculus of tensors Tensoranalysis Einführung |
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