Tensor analysis:
Saved in:
| Main Authors: | , |
|---|---|
| Other Authors: | , |
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Boston
De Gruyter
[2018]
|
| Series: | De Gruyter Graduate
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 327 Seiten Illustrationen, Diagramme |
| ISBN: | 9783110404258 |
Staff View
MARC
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| 245 | 1 | 0 | |a Tensor analysis |c Heinz Schade, Klaus Neemann ; translated by Andrea Dziubek and Edmond Rusjan |
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Record in the Search Index
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| adam_text | CONTENTS
PREFACE * V
ACKNOWLEDGMENTS OF THE TRANSLATORS * IX
1 ALGEBRAIC TOOLS * 1
1.1 THE SUMMATION CONVENTION * 1
1.2 N-TUPLES * 4
1.2.1 DEFINITIONS * 4
1.2.2 LINEAR OPERATIONS * 4
1.2.3 LINEAR INDEPENDENCE * 5
1.3 DETERMINANTS * 6
1.3.1 DEFINITIONS * 6
1.3.2 COMPUTING DETERMINANTS
----
7
1.3.3 COMPUTATIONS WITH DETERMINANTS * 8
1.4 KRONECKER SYMBOLS * 10
1.4.1 OIJ * 10
1.4.2 OEPIQ
----
11
1.4.3 /.../
----
12
1.4.4 REPRESENTATION OF A DETERMINANT BY E, / * 15
1.4.5 EIJK * 18
1.5 MATRICES * 20
1.5.1 DEFINITIONS * 20
1.5.2 ARITHMETIC OPERATIONS AND FIRST CONCLUSIONS * 21
1.5.3 EQUATIONS INVOLVING MATRICES AND EQUATIONS INVOLVING MATRIX
ELEMENTS * 25
1.5.4 ELEMENTARY TRANSFORMATIONS, NORMAL FORM, EQUIVALENT MATRICES,
SIMILAR MATRICES * 26
1.5.5 ORTHOGONAL MATRICES * 28
1.6 ALGORITHMS * 29
1.6.1 COMPUTING THE VALUE OF A DETERMINANT * 29
1.6.2 SOLUTION OF SYSTEMS OF LINEAR EQUATIONS WITH THE SAME REGULAR
COEFFICIENT MATRIX (*DIVISION BY A REGULAR MATRIX*, GAUSS-JORDAN
ALGORITHM)
----
29
1.6.3 DETERMINING THE RANK OF A MATRIX OR A DETERMINANT * 30
2 TENSOR ANALYSIS IN SYMBOLIC NOTATION AND IN CARTESIAN COORDINATES *
2.1 CARTESIAN COORDINATES, POINTS, POSITION VECTORS * 31
2.1.1 POSITION VECTORS AND POINT COORDINATES * 31
2.1.2 TRANSFORMATIONS OF CARTESIAN COORDINATE SYSTEMS * 31
2.1.3
2.1.4
2.1.5
2.2
2
.
2.1
2
.
2.2
2.3
2.3.1
2.3.2
2.3.3
2.4
2.5
2.6
2.7
2.7.1
2.7.2
2.7.3
2.7.4
2.8
2
.
8.1
2
.
8.2
2.8.3
2.9
2.9.1
2.9.2
2.9.3
2.9.4
2.10
2
.
10.1
2
.
10.2
2.10.3
2.11
2.12
2
.
12.1
2
.
12.2
2.12.3
2.12.4
2.12.5
2
.
12.6
PROPERTIES OF THE TRANSFORMATION COEFFICIENTS * 32
TRANSFORMATION LAW FOR BASIS VECTORS * 34
TRANSFORMATION LAW FOR POINT COORDINATES * 35
VECTORS * 35
VECTORS, VECTOR COMPONENTS, AND VECTOR COORDINATES * 35
THE TRANSFORMATION LAW FOR VECTOR COORDINATES * 36
TENSORS * 40
SECOND-ORDER TENSORS * 40
TENSORS OF ARBITRARY ORDER * 43
SYMMETRIES IN PHYSICS * 44
SYMBOLIC NOTATION, INDEX NOTATION, AND MATRIX NOTATION * 45
EQUALITY, ADDITION, AND SUBTRACTION OF TENSORS.
MULTIPLICATION OF A TENSOR BY A SCALAR.
LINEAR INDEPENDENCE * 46
TRANSPOSED, ISOMETRIC, SYMMETRIC, AND ANTIMETRIC TENSORS * 48
TENSOR MULTIPLICATION OF TENSORS * 50
DEFINITION
----
50
PROPERTIES * 51
TENSORS, TENSOR COMPONENTS, AND TENSOR COORDINATES * 54
TENSOR EQUATIONS, TRANSFORMATION EQUATIONS, AND REPRESENTATION
EQUATIONS
----
55
5-TENSOR, E-TENSOR, ISOTROPIC TENSORS * 55
THE 5-TENSOR * 55
THE E-TENSOR * 56
ISOTROPIC TENSORS
----
57
SCALAR MULTIPLICATION OF TENSORS * 57
DEFINITION * 57
PROPERTIES * 58
CONTRACTION AND TRACE * 63
MULTIPLE SCALAR PRODUCTS * 63
VECTOR MULTIPLICATION OF TENSORS * 65
DEFINITION * 65
PROPERTIES * 68
TRIPLE PRODUCT
----
69
SUMMARY OF TENSOR-ALGEBRAIC OPERATIONS * 70
DIFFERENTIAL OPERATIONS * 71
THE FUNDAMENTAL THEOREM OF TENSOR ANALYSIS * 71
GRADIENT * 71
(TOTAL) DIFFERENTIAL * 74
DIVERGENCE * 76
CURL * 77
LAPLACE OPERATOR * 79
2.13 RULES FOR INDICES AND UNDERLINES * 79
2.14 INTEGRALS OF TENSOR FIELDS * 80
2.14.1 LINE INTEGRALS OF TENSOR COORDINATES * 80
2.14.2 NORMAL VECTOR AND SURFACE VECTOR OF A SURFACE ELEMENT * 82
2.14.3 SURFACE INTEGRALS OF TENSOR COORDINATES * 85
2.14.4 VOLUME INTEGRALS OF TENSOR COORDINATES * 88
2.14.5 INTEGRALS OF HIGHER-ORDER TENSOR FIELDS * 90
2.15 GAUSS THEOREM AND STOKES THEOREM * - 91
2.15.1 GAUSS THEOREM * 91
2.15.2 STOKES THEOREM * 94
3 ALGEBRA OF SECOND-ORDER TENSORS * 99
3.1 ADDITIVE DECOMPOSITION OF A TENSOR * 99
3.2 THE DETERMINANT OF A TENSOR * 100
3.3 THE VECTOR ASSOCIATED WITH AN ANTIMETRIC TENSOR* 101
3.4 THE COTENSOR OF A TENSOR * 102
3.5 THE RANK OF A TENSOR* 102
3.6 THE INVERSE TENSOR * 103
3.7 ORTHOGONAL TENSORS * 104
3.8 TENSORS AS LINEAR VECTOR FUNCTIONS * 105
3.8.1 RANK 3 * 106
3.8.2 RANK 2 * 107
3.8.3 R ANKL
----
108
3.8.4 RANKO * 109
3.9 RECIPROCAL BASES * 109
3.9.1 DEFINITION * 109
3.9.2 ORTHOGONALITY RELATIONS * 110
3.9.3 ORTHOGONAL BASES AND ORTHONORMAL BASES * 111
3.9.4 RECIPROCAL BASES IN THE PLANE * 111
3.10 REPRESENTATION OF A TENSOR BY VECTORS * 112
3.10.1 RANK 3 * 113
3.10.2 RANK 2 * 114
3.10.3 RANKL * 116
3.11 EIGENVALUES AND EIGENVECTORS. THE CHARACTERISTIC EQUATION * 117
3.11.1 EIGENVALUES AND EIGENVECTORS * 117
3.11.2 THE CHARACTERISTIC EQUATION AND MAIN INVARIANTS * 118
3.11.3 CLASSIFICATION OF TENSORS BY THE TYPE OF THEIR EIGENVALUES,
EIGENVALUE THEOREMS * 119
3.11.4 THEOREMS ABOUT EIGENVECTORS * 122
3.11.5 EIGENVALUES AND EIGENVECTORS OF SQUARE MATRICES * 128
3.12 SYMMETRIC TENSORS * 130
3.12.1 PRINCIPAL AXES TRANSFORMATION * 130
3.12.2
3.12.3
3.12.4
3.13
3.13.1
3.13.2
3.13.3
3.13.4
3.14
3.14.1
3.14.2
3.14.3
3.15
3.16
4
4.1
4.1.1
4.1.2
4.1.3
4.1.4
4.1.5
4.2
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
4.2.6
4.2.7
4.2.8
4.3
4.4
4.4.1
4.4.2
4.4.3
4.4.4
4.4.5
EIGENVALUES AND RANK OF A TENSOR* 135
EIGENVALUES AND DEFINITENESS OF A TENSOR * 135
SYMMETRIC MATRICES * 137
ORTHOGONAL POLAR TENSORS * 140
ROTATION IN A PLANE * 140
TRANSFORMATION TO AN EIGENDIRECTION * 140
THE ORTHOGONAL TENSOR AS A FUNCTION OF ROTATION ANGLE AND ROTATION
AXIS OR REFLECTION AXIS
----
144
ROTATION AND COORDINATE TRANSFORMATION * 148
POWERS OF TENSORS. CAYLEY-HAMILTON THEOREM * 149
POWERS WITH INTEGER EXPONENTS * 149
POWERS WITH REAL EXPONENTS * 150
CAYLEY-HAMILTON THEOREM * 152
BASIC INVARIANTS * 153
POLAR DECOMPOSITION OF A TENSOR
----
154
TENSOR ANALYSIS IN CURVILINEAR COORDINATES * 159
CURVILINEAR COORDINATES * 159
CURVILINEAR COORDINATE SYSTEMS * 159
COORDINATE SURFACES AND COORDINATE CURVES * 161
HOLONOMIC BASES * 162
RECTILINEAR AND CARTESIAN COORDINATE SYSTEMS * 164
ORTHOGONAL COORDINATE SYSTEMS
----
165
HOLONOMIC TENSOR COORDINATES * 165
INTRODUCTION * 165
TRANSFORMATIONS BETWEEN TWO CURVILINEAR COORDINATE SYSTEMS * 168
THE SUMMATION CONVENTION * 170
THE D-TENSOR* 171
RAISING AND LOWERING INDICES * 175
THE E-TENSOR* 176
ISOTROPIC TENSORS * 179
TENSOR ALGEBRA IN HOLONOMIC COORDINATES
----
179
PHYSICAL BASES AND TENSOR COORDINATES * 184
TENSOR-ANALYTICAL OPERATIONS * 186
PARTIAL DERIVATIVE AND DIFFERENTIAL OF THE POSITION VECTOR * 187
PARTIAL DERIVATIVE AND TOTAL DIFFERENTIAL OF THE HOLONOMIC BASES,
CHRISTOFFEL SYMBOLS * 187
CHRISTOFFEL SYMBOLS AND METRIC COEFFICIENTS * 188
PARTIAL DERIVATIVES OF TENSORS. PARTIAL AND COVARIANT DERIVATIVES OF
TENSOR COORDINATES * 190
THE TOTAL DIFFERENTIAL OF TENSORS. THE TOTAL AND THE ABSOLUTE
DIFFERENTIAL
OF TENSOR COORDINATES * 191
4.4.6 DERIVATIVES WITH RESPECT TO A PARAMETER* 193
4.4.7 GRADIENT * 193
4.4.8 DIVERGENCE AND CURL * 194
4.4.9 PHYSICAL COORDINATES OF TENSOR-ANALYTICAL OPERATIONS * 195
4.4.10 SECOND COVARIANT DERIVATIVE OF A TENSOR COORDINATE.
LAPLACE OPERATOR
----
197
4.4.11 INTEGRALS OF TENSOR FIELDS * 199
4.5 FUNDAMENTALS OF THE THEORY OF SURFACES * 202
4.5.1 SURFACE COORDINATES AND MOVING FRAMES * 203
4.5.2 SURFACE VECTORS AND SURFACE TENSORS * 205
4.5.3 METRIC PROPERTIES * 206
4.5.4 CURVATURE PROPERTIES
----
210
4.5.5 DERIVATIVE EQUATIONS, INTEGRABILITY CONDITIONS * 215
5 REPRESENTATION OF TENSOR FUNCTIONS * 221
5.1 THE BASIC IDEA OF THE REPRESENTATION OF TENSOR FUNCTIONS * 221
5.2 GENERALIZED CAYLEY-HAMILTON THEOREM * 223
5.3 INVARIANTS OF VECTORS AND SECOND-ORDER TENSORS * 225
5.3.1 INVARIANTS OF VECTORS * 226
5.3.2 INDEPENDENT INVARIANTS OF A SECOND-ORDER TENSOR* 226
5.3.3 IRREDUCIBLE INVARIANTS OF SECOND-ORDER TENSORS * 228
5.3.4 SUMMARY
----
236
5.4 ISOTROPIC TENSOR FUNCTIONS * 236
5.4.1 INVARIANCE CONDITIONS * 236
5.4.2 SCALAR-VALUED FUNCTIONS * 238
5.4.3 VECTOR-VALUED FUNCTIONS * 238
5.4.4 TENSOR-VALUED FUNCTIONS * 240
5.4.5 SUMMARY * 243
5.5 CONSIDERING ANISOTROPY * 244
REFERENCES * 249
APPENDICES * 253
A SOLUTIONS TO THE PROBLEMS * 253
B CYLINDRICAL COORDINATES AND SPHERICAL COORDINATES * 309
B.L CYLINDRICAL COORDINATES * 309
B.1.1 TRANSFORMATION EQUATIONS FOR POINT COORDINATES * 309
B.L.2 BASES * 310
B.1.3 TRANSFORMATION EQUATIONS FOR TENSOR COORDINATES * 310
B.1.4 OE-TENSOR AND E-TENSOR * 311
B .L.5
B.1.6
B .L.7
B.2
B.2.1
B.2.2
B.2.3
B.2.4
B.2.5
B.2.6
B.2.7
CHRISTOFFEL SYMBOLS * 312
DIFFERENTIAL OPERATORS * 312
CURVE-, AREA-, AND VOLUME ELEMENTS * 313
SPHERICAL COORDINATES
----
314
TRANSFORMATION EQUATIONS FOR POINT COORDINATES -
BASES * 315
TRANSFORMATION EQUATIONS FOR TENSOR COORDINATES
OE-TENSOR AND E-TENSOR * 318
CHRISTOFFEL SYMBOLS * 318
DIFFERENTIAL OPERATORS * 319
CURVE-, AREA-, AND VOLUME ELEMENTS * 320
INDEX * 321
|
| any_adam_object | 1 |
| author | Schade, Heinz 1933- Neemann, Klaus |
| author2 | Dziubek, Andrea Rusjan, Edmond |
| author2_role | trl trl |
| author2_variant | a d ad e r er |
| author_GND | (DE-588)129335576 (DE-588)121017788 |
| author_facet | Schade, Heinz 1933- Neemann, Klaus Dziubek, Andrea Rusjan, Edmond |
| author_role | aut aut |
| author_sort | Schade, Heinz 1933- |
| author_variant | h s hs k n kn |
| building | Verbundindex |
| bvnumber | BV043976280 |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)921524986 (DE-599)DNB1076282563 |
| dewey-full | 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510 |
| dewey-search | 510 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre_facet | Lehrbuch |
| id | DE-604.BV043976280 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:40:10Z |
| institution | BVB |
| isbn | 9783110404258 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029384763 |
| oclc_num | 921524986 |
| open_access_boolean | |
| owner | DE-703 DE-19 DE-BY-UBM DE-1050 DE-29T DE-83 DE-706 DE-573 DE-20 DE-11 DE-188 |
| owner_facet | DE-703 DE-19 DE-BY-UBM DE-1050 DE-29T DE-83 DE-706 DE-573 DE-20 DE-11 DE-188 |
| physical | XVI, 327 Seiten Illustrationen, Diagramme |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | De Gruyter |
| record_format | marc |
| series2 | De Gruyter Graduate |
| spelling | Schade, Heinz 1933- Verfasser (DE-588)129335576 aut Tensoranalysis (1997) Tensor analysis Heinz Schade, Klaus Neemann ; translated by Andrea Dziubek and Edmond Rusjan Berlin ; Boston De Gruyter [2018] © 2018 XVI, 327 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier De Gruyter Graduate Tensoranalysis (DE-588)4204323-2 gnd rswk-swf Tensor (DE-588)4184723-4 gnd rswk-swf Fachhochschul-/Hochschulausbildung (DE-588)4123623-3 Lehrbuch gnd-content Tensoranalysis (DE-588)4204323-2 s DE-604 Tensor (DE-588)4184723-4 s 1\p DE-604 Neemann, Klaus Verfasser (DE-588)121017788 aut Dziubek, Andrea trl Rusjan, Edmond trl Erscheint auch als Online-Ausgabe, PDF 978-3-11-040426-5 Erscheint auch als Online-Ausgabe, EPUB 978-3-11-040549-1 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029384763&sequence=000001&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 | Schade, Heinz 1933- Neemann, Klaus Tensor analysis Tensoranalysis (DE-588)4204323-2 gnd Tensor (DE-588)4184723-4 gnd |
| subject_GND | (DE-588)4204323-2 (DE-588)4184723-4 (DE-588)4123623-3 |
| title | Tensor analysis |
| title_alt | Tensoranalysis (1997) |
| title_auth | Tensor analysis |
| title_exact_search | Tensor analysis |
| title_full | Tensor analysis Heinz Schade, Klaus Neemann ; translated by Andrea Dziubek and Edmond Rusjan |
| title_fullStr | Tensor analysis Heinz Schade, Klaus Neemann ; translated by Andrea Dziubek and Edmond Rusjan |
| title_full_unstemmed | Tensor analysis Heinz Schade, Klaus Neemann ; translated by Andrea Dziubek and Edmond Rusjan |
| title_short | Tensor analysis |
| title_sort | tensor analysis |
| topic | Tensoranalysis (DE-588)4204323-2 gnd Tensor (DE-588)4184723-4 gnd |
| topic_facet | Tensoranalysis Tensor Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029384763&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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