From vectors to tensors:
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| Main Authors: | , |
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| Language: | English |
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2005
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| Series: | Universitext
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| Item Description: | Auch als Internetausgabe |
| Physical Description: | XVI, 670 Seiten Diagramme |
| ISBN: | 354022887X |
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| 020 | |a 354022887X |c Pb. : EUR 64.15 (freier Pr.), ca. sfr 106.00 (freier Pr.) |9 3-540-22887-X | ||
| 024 | 3 | |a 9783540228875 | |
| 028 | 5 | 2 | |a 11312567 |
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| 100 | 1 | |a Ruíz-Tolosa, Juan Ramón |0 (DE-588)129615714 |4 aut | |
| 245 | 1 | 0 | |a From vectors to tensors |c Juan Ramón Ruíz-Tolosa ; Enrique Castillo |
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c 2005 | |
| 300 | |a XVI, 670 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Universitext | |
| 500 | |a Auch als Internetausgabe | ||
| 650 | 4 | |a Algèbre tensorielle | |
| 650 | 4 | |a Algèbre vectorielle | |
| 650 | 4 | |a Analyse vectorielle | |
| 650 | 4 | |a Calcul tensoriel | |
| 650 | 4 | |a Calculus of tensors |v Textbooks | |
| 650 | 4 | |a Tensor algebra |v Textbooks | |
| 650 | 4 | |a Vector algebra |v Textbooks | |
| 650 | 0 | 7 | |a Tensoralgebra |0 (DE-588)4505278-5 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
| 689 | 0 | 0 | |a Tensoralgebra |0 (DE-588)4505278-5 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Castillo, Enrique |d 1946- |0 (DE-588)124179266 |4 aut | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-013029913 | |
Record in the Search Index
| _version_ | 1814170434061467648 |
|---|---|
| adam_text |
Contents
Part I Basic Tensor Algebra
Tensor Spaces
. 3
1.1
Introduction
. 3
1.2
Dual or reciprocal coordinate frames in
affine
Euclidean spaces
3
1.3
Different types of matrix products
. 8
1.3.1
Definitions
. 8
1.3.2
Properties concerning general matrices
. 10
1.3.3
Properties concerning square matrices
. 11
1.3.4
Properties concerning eigenvalues and eigenvectors
. 12
1.3.5
Properties concerning the
Schur
product
. 13
1.3.6
Extension and condensation of matrices
. 13
1.3.7
Some important matrix equations
. 17
1.4
Special tensors
. 26
1.5
Exercises
. 30
Introduction to Tensors
. 33
2.1
Introduction
. 33
2.2
The triple tensor product linear space
. . 33
2.3
Einstein's summation convention
. 36
2.4
Tensor analytical representation
. 37
2.5
Tensor product axiomatic properties
. 38
2.6
Generalization
. 40
2.7
Illustrative examples
. 41
2.8
Exercises
. 46
Homogeneous Tensors
. 47
3.1
Introduction
. 47
3.2
The concept of homogeneous tensors
. 47
3.3
General rules about tensor notation
. 48
3.4
The tensor product of tensors
. 50
X
Contents
3.5
Einstein's contraction of the tensor product
. 54
3.6
Matrix representation of tensors
. 56
3.6.1
First-order tensors
. 56
3.6.2
Second-order tensors
. 57
3.7
Exercises
. 61
4
Change-of-basis in Tensor Spaces
. 65
4.1
Introduction
. 65
4.2
Change of basis in a third-order tensor product space
. 65
4.3
Matrix representation of a change-of-basis in tensor spaces
. 67
4.4
General criteria for tensor character
. 69
4.5
Extension to homogeneous tensors
. 72
4.6
Matrix operation rules for tensor expressions
. 74
4.6.1
Second-order tensors (matrices)
. 74
4.6.2
Third-order tensors
. 77
4.6.3
Fourth-order tensors
. 78
4.7
Change-of-basis invariant tensors:
Isotropie
tensors
. 80
4.8
Main
isotropie
tensors
. 80
4.8.1
The null tensor
. 80
4.8.2
Zero-order tensor (scalar invariant)
. 80
4.8.3 Kronecker'
s
delta
. 80
4.9
Exercises
. 106
5
Homogeneous Tensor Algebra: Tensor Homomorphisms
.
Ill
5.1
Introduction
.
Ill
5.2
Л4аіп
theorem on tensor contraction
.
Ill
5.3
The contracted tensor product and tensor homomorphisms
. 113
5.4
Tensor product applications
.119
5.4.1
Common simply contracted tensor products
.119
5.4.2
Multiply contracted tensor products
.120
5.4.3
Scalar and inner tensor products
.120
5.5
Criteria for tensor character based on contraction
.122
5.6
The contracted tensor product in the reverse sense: The
quotient law
.124
5.7
Matrix representation of permutation homomorphisms
.127
5.7.1
Permutation matrix tensor product types in Kn
.127
5.7.2
Linear span of precedent types
.129
5.7.3
The isomers of a tensor
.137
5.8
Matrices associated with simply contraction homomorphisms
. 141
5.8.1
Mixed tensors of second order (r
= 2):
Matrices
.141
5.8.2 ,
Mixed tensors of third order (r
= 3).141
5.8.3
Mixed tensors of fourth order (r
= 4) .142
5.8.4
Mixed tensors of fifth order (r
= 5) .143
5.9
Matrices associated with doubly contracted homomorphisms
. 144
5.9.1
Mixed tensors of fourth order (r
— 4) .144
Contents
XI
5.9.2
Mixed tensors of fifth order (r
= 5) .145
5.10 Eigentensors.159
5.11
Generalized multilinear mappings
.165
5.11.1
Theorems of similitude with tensor mappings
.167
5.11.2
Tensor mapping types
.168
5.11.3
Direct
n-dimensional
tensor endomorphisms
.169
5.12
Exercises
.183
Part II Special Tensors
6
Symmetric Homogeneous Tensors: Tensor Algebras
.189
6.1
Introduction
.189
6.2
Symmetric systems of scalar components
.189
6.2.1
Symmetric systems with respect to an index subset
. . 190
6.2.2
Symmetric systems. Total symmetry
.190
6.3
Strict components of a symmetric system
.191
6.3.1
Number of strict components of a symmetric system
with respect to an index subset
.191
6.3.2
Number of strict components of a symmetric system
. 192
6.4
Tensors with symmetries: Tensors with branched symmetry,
symmetric tensors
.193
6.4.1
Generation of symmetric tensors
.194
6.4.2
Intrinsic character of tensor S3rmmetry: Fundamental
theorem of tensors with symmetry
.197
6.4.3
Symmetric tensor spaces and subspaces. Strict
components associated with subspaces
.204
6.5
Symmetric tensors under the tensor algebra, perspective
.206
6.5.1
Symmetrized tensor associated with an arbitrary
pure tensor
.210
6.5.2
Extension of the symmetrized tensor associated with
a mixed tensor
.210
6.6
Symmetric tensor algebras: The &s product
.212
6.7
Illustrative examples
.214
6.8
Exercises
.220
7
Anti-symmetric Homogeneous Tensors, Tensor and Inner
Product Algebras
.225
7.1
Introduction
.225
7.2
Anti-symmetric systems of scalar components
.225
7.2.1
Anti-symmetric systems with respect to an index
subset
.226
7.2.2
Anti-symmetric systems. Total anti-symmetry
.228
7.3
Strict components of an anti-symmetric system and with
respect to an index subset
.228
XII Contents
7.3.1
Number of strict components of an anti-symmetric
system with respect to an index subset
.229
7.3.2
Number of strict components of an anti-symmetric
system
. 229
.7.4
Tensors with anti-symmetries: Tensors with branched
anti-symmetry; anti-symmetric tensors
.230
7.4.1
Generation of anti-symmetric tensors
.232
7.4.2
Intrinsic character of tensor anti-symmetry:
Fundamental theorem of tensors with anti-symmetry
. 236
7.4.3
Anti-symmetric tensor spaces and subspaces. Vector
subspaces associated with strict components
.243
7.5
Anti-symmetric tensors from the tensor algebra perspective
. . 246
7.5.1
Anti-symmetrized tensor associated with an
arbitrary pure tensor
. 249
7.5.2
Extension of the anti-symmetrized tensor concept
associated with a mixed tensor
.249
7.6
Anti-symmetric tensor algebras: The
®я
product
.252
7.7
Illustrative examples
.253
7.8
Exercises
.265
8
Pseudotenşors;
Modular, Relative or Weighted Tensors
.269
8.1
Introduction
.269
8.2
Previous concepts of modular tensor establishment
.269
8.2.1
Relative modulus of a change-of-basis
.269
8.2.2
Oriented vector space
.270
8.2.3
Weight tensor
.270
8.3
Axiomatic properties for the modular tensor concept
.270
8.4
Modular tensor characteristics
.271
8.4.1
Equality of modular tensors
.272
8.4.2
Classification and special denominations
.272
8.5
Remarks on modular tensor operations: Consequences
.272
8.5.1
Tensor addition
.272
8.5.2
Multiplication by a scalar
.274
8.5.3
Tensor product
.275
8.5.4 ·
Tensor contraction
.276
8.5.5
Contracted tensor products
.276
8.5.6
The quotient law. New criteria for modular tensor
character
.277
8.6
Modular symmetry and anti-symmetry
.280
8.7
Main modular tensors
.291
8.7.1
e
systems, permutation systems or Levi-Civita
•
tensor systems
.291
8.7.2
Generalized
Kronecker
deltas: Definition
.293
8.7.3
Dual or polar tensors: Definition
.301
8.8
Exercises
.310
Part III Exterior Algebras
9
Exterior Algebras:
Totally Anti-symmetric Homogeneous Tensor Algebras
.315
9.1
Introduction and Definitions
.315
9.1.1
Exterior product of two vectors
.315
9.1.2
Exterior product of three vectors
.317
9.1.3*
Strict components of exterior vectors.
M
ulti
vectors
. . . 318
9.2
Exterior product of
r
vectors: Decomposable multivectors
. 319
9.2.1
Properties of exterior products of order r:
Decomposable multivectors or exterior vectors
.321
9.2.2
Exterior algebras over Vn(K) spaces: Terminology
. . . 323
9.2.3
Exterior algebras of order r=0 and r=l
.324
9.3
Axiomatic properties of tensor operations in exterior algebras
324
9.3.1
Addition and multiplication by an scalar
.324
9.3.2
Generalized exterior tensor product: Exterior
product of exterior vectors
.325
9.3.3
Anti-commutativity of the exterior product
Д
. . 331
9.4
Dual exterior algebras over V^(K) spaces
.331
9.4.1
Exterior product of
r
linear forms over V™(K)
.332
9.4.2
Axiomatic tensor operations in dual exterior
Algebras
/\n¿{K)-
Dual exterior tensor product
.333
9.4.3
Observation about bases of primary and dual
exterior spaces
.,.334
9.5
The change-of-basis in exterior algebras
.337
9.5.1
Strict tensor relationships for
/\£ (K)
algebras
.338
9.5.2
Strict tensor relationships for
f\„}(R)
algebras
.339
9.6
Complements of
contramodular
and comodular scalars
.341
9.7
Comparative tables of algebra correspondences
.342
9.8
Scalar mappings: Exterior contractions
.342
9.9
Exterior vector mappings: Exterior homornorphisms
.345
9.9.1
Direct exterior endomorphism
.350
9.10
Exercises
.383
10
Mixed Exterior Algebras
.387
10.1
Introduction
.387
10.1.1
Mixed anti-symmetric tensor spaces and their strict
tensor components
.387
10.1.2
Mixed exterior product of four vectors
.390
10.2
Decomposable mixed exterior vectors
.394
10.3
Mixed exterior algebras: Terminology
.397
10.3.1
Exterior basis of a mixed exterior algebra
.397
10.3.2
Axiomatic tensor operations in the f\^ {K) algebra
398
XIV Contents
10.4
Exterior product
of mixed exterior vectors
.399
10.5
Anti-commutativity of the
Д
mixed exterior product
.403
10.6
Change of basis in mixed exterior algebras
.404
10.7
Exercises
.409
Part IV Tensors over Linear Spaces with Inner Product
11
Euclidean Homogeneous Tensors
.413
11.1
Introduction
.413
11.2
Initial concepts
.413
11.3
Tensor character of the inner vector's connection in a
PSEń(U.)
space
.416
11.4
Different types of the fundamental connection tensor
.418
11.5
Tensor product of vectors in
Еп(П)
(or in PSEn{U))
.421
11.6
Equivalent associated tensors: Vertical displacements of
indices. Generalization
.422
11.6.1
The quotient space of isomers
.426
11.7
Changing bases in
Еп(Л):
Euclidean tensor character criteria
427
11.8
Symmetry and anti-symmetry in Euclidean tensors
.430
11.9
Cartesian tensors
.433
11.9.1
Main properties of Euclidean
En{H)
spaces in
orthonormal
bases
.433
11.9.2
Tensor total Euclidean character in
orthonormal
bases
.434
11.9.3.
Tensor partial Euclidean character in
orthonormal
bases
.436
11.9.4
Rectangular Cartesian tensors
.436
11.10
Euclidean and pseudo-Euclidean tensor algebra
.451
11.10.1
Euclidean tensor equality
.451
11.10.2
Addition and external product of Euclidean
(pseudo-Euclidean) tensors
.451
11.10.3
Tensor product of Euclidean (pseudo-Euclidean)
tensors
. 452
11.10.4
Euclidean (pseudo-Euclidean) tensor contraction
.452
11.10.5
Contracted tensor product of Euclidean or
pseudo-Euclidean tensors
.455
11.10.6
Euclidean contraction of tensors of order
r
= 2.457
11.10.7
Euclidean contraction of tensors of order
r
= 3.457
11.10.8
Euclidean contraction of tensors of order
r
= 4.457
11.10.9
Euclidean contraction of indices by the
Hadamard
product
.458
11.11
Euclidean tensor metrics
.482
11.11.1
Inner connection
.483
11.11.2
The induced fundamental metric tensor
.484
Contents
XV
11.11.3
Reciprocai
and
orthonormal
basis
.486
11.12
Exercises
.504
12
Modular Tensors over En(lR) Euclidean Spaces
.511
12.1
Introduction
.511
12.2
Diverse cases of linear space connections
.511
12.3
Tensor character of
γ
/jÔj
.
512
12.4
The orientation tensor: Definition
.514
12.5
Tensor character of the orientation tensor
.514
12.6
Orientation tensors as associated Euclidean tensors
.515
12.7
Dual or polar tensors over En(lR) Euclidean spaces
.516
12.8
Exercises
.525
13
Euclidean Exterior Algebra
.529
13.1
Introduction
.529
13.2
Euclidean exterior algebra of order
τ
— 1 .529
13.3
Euclidean exterior algebra of order
r
(2 <
г
<
n)
.532
13.4
Euclidean exterior algebra of order
r—n
. 535
13:5
The orientation tensor in exterior bases
.535
13.6
Dual or polar tensors in exterior bases
.536
13.7
The cross product as a polar tensor in generalized Cartesian
coordinate frames
.538
13.8
\АЩ
geometric interpretation in generalized Cartesian
coordinate frames
.539
13.9
Illustrative examples
.540
13.10
Exercises
.576
Part V Classic Tensors in Geometry and Mechanics
14
Affine
Tensors
.581
14.1
Introduction and Motivation
.581
14.2
Euclidean tensors in En(R)
.582
14.2.1
Projection tensor
.582
14.2.2
The momentum tensor
.586
14.2.3
The rotation tensor
.587
14.2.4
The reflection tensor
.590
14.3
Affine
geometric tensors.
Homographies
.597
14.3.1
Preamble
.597
14.3.2
Definition and representation
.599
14.3.3
Affinities
.600
14.3.4-
Homothecies
_.604
14.3.5
Isometries
.606
14.3.6
Product of isometries
.623
14.4
Tensors in Physics and Mechanics
.626
XVI Contents
14.4.1
The stress tensor
5.628
14.4.2
The strain tensor
Γ
. 630
14.4.3
Tensor relationships between
S
and
Γ.
Elastic tensor.
635
14.4.4
The
inerţial
moment tensor
.647
14.5
Exercises
.655
Bibliography
.659
Index
.663 |
| any_adam_object | 1 |
| author | Ruíz-Tolosa, Juan Ramón Castillo, Enrique 1946- |
| author_GND | (DE-588)129615714 (DE-588)124179266 |
| author_facet | Ruíz-Tolosa, Juan Ramón Castillo, Enrique 1946- |
| author_role | aut aut |
| author_sort | Ruíz-Tolosa, Juan Ramón |
| author_variant | j r r t jrr jrrt e c ec |
| building | Verbundindex |
| bvnumber | BV019702378 |
| callnumber-first | Q - Science |
| callnumber-label | QA433 |
| callnumber-raw | QA433 |
| callnumber-search | QA433 |
| callnumber-sort | QA 3433 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 220 SK 350 |
| classification_tum | MAT 157f |
| ctrlnum | (OCoLC)57333271 (DE-599)BVBBV019702378 |
| dewey-full | 515/.63 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.63 |
| dewey-search | 515/.63 |
| dewey-sort | 3515 263 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV019702378 |
| illustrated | Not Illustrated |
| indexdate | 2024-10-28T15:02:28Z |
| institution | BVB |
| isbn | 354022887X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013029913 |
| oclc_num | 57333271 |
| open_access_boolean | |
| owner | DE-703 DE-91 DE-BY-TUM DE-824 DE-384 DE-526 DE-634 DE-20 DE-11 DE-188 DE-83 |
| owner_facet | DE-703 DE-91 DE-BY-TUM DE-824 DE-384 DE-526 DE-634 DE-20 DE-11 DE-188 DE-83 |
| physical | XVI, 670 Seiten Diagramme |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Ruíz-Tolosa, Juan Ramón (DE-588)129615714 aut From vectors to tensors Juan Ramón Ruíz-Tolosa ; Enrique Castillo Berlin ; Heidelberg Springer 2005 XVI, 670 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Universitext Auch als Internetausgabe Algèbre tensorielle Algèbre vectorielle Analyse vectorielle Calcul tensoriel Calculus of tensors Textbooks Tensor algebra Textbooks Vector algebra Textbooks Tensoralgebra (DE-588)4505278-5 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Tensoralgebra (DE-588)4505278-5 s DE-604 Castillo, Enrique 1946- (DE-588)124179266 aut Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013029913&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ruíz-Tolosa, Juan Ramón Castillo, Enrique 1946- From vectors to tensors Algèbre tensorielle Algèbre vectorielle Analyse vectorielle Calcul tensoriel Calculus of tensors Textbooks Tensor algebra Textbooks Vector algebra Textbooks Tensoralgebra (DE-588)4505278-5 gnd |
| subject_GND | (DE-588)4505278-5 (DE-588)4123623-3 |
| title | From vectors to tensors |
| title_auth | From vectors to tensors |
| title_exact_search | From vectors to tensors |
| title_full | From vectors to tensors Juan Ramón Ruíz-Tolosa ; Enrique Castillo |
| title_fullStr | From vectors to tensors Juan Ramón Ruíz-Tolosa ; Enrique Castillo |
| title_full_unstemmed | From vectors to tensors Juan Ramón Ruíz-Tolosa ; Enrique Castillo |
| title_short | From vectors to tensors |
| title_sort | from vectors to tensors |
| topic | Algèbre tensorielle Algèbre vectorielle Analyse vectorielle Calcul tensoriel Calculus of tensors Textbooks Tensor algebra Textbooks Vector algebra Textbooks Tensoralgebra (DE-588)4505278-5 gnd |
| topic_facet | Algèbre tensorielle Algèbre vectorielle Analyse vectorielle Calcul tensoriel Calculus of tensors Textbooks Tensor algebra Textbooks Vector algebra Textbooks Tensoralgebra Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013029913&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT ruiztolosajuanramon fromvectorstotensors AT castilloenrique fromvectorstotensors |