From Vectors to Tensors

From Vectors to Tensors [Ruiz-Tolosa, Castillo]......Page 12082463_0001.djvu
Title......Page all_16951_to_0225.cpc0002.djvu
Copyright......Page all_16951_to_0225.cpc0003.djvu
Dedication......Page all_16951_to_0225.cpc0004.djvu
Preface......Page all_16951_to_0225.cpc0005.djvu
Contents......Page all_16951_to_0225.cpc0007.djvu
Part I - Basic Tensor Algebra......Page all_16951_to_0225.cpc0015.djvu
1.2 Dual or reciprocal coordinate frames in affine Euclidean spaces......Page all_16951_to_0225.cpc0016.djvu
1.3 Different types of matrix products......Page all_16951_to_0225.cpc0021.djvu
1.4 Special tensors......Page all_16951_to_0225.cpc0039.djvu
1.5 Exercises......Page all_16951_to_0225.cpc0043.djvu
2.2 The triple tensor product linear space......Page all_16951_to_0225.cpc0045.djvu
2.3 Einstein\'s summation convention......Page all_16951_to_0225.cpc0048.djvu
2.4 Tensor analytical representation......Page all_16951_to_0225.cpc0049.djvu
2.5 Tensor product axiomatic properties......Page all_16951_to_0225.cpc0050.djvu
2.6 Generalization......Page all_16951_to_0225.cpc0052.djvu
2.7 Illustrative examples......Page all_16951_to_0225.cpc0053.djvu
2.8 Exercises......Page all_16951_to_0225.cpc0058.djvu
3.2 The concept of homogeneous tensors......Page all_16951_to_0225.cpc0059.djvu
3.3 General rules about tensor notation......Page all_16951_to_0225.cpc0060.djvu
3.4 The tensor product of tensors......Page all_16951_to_0225.cpc0062.djvu
3.5 Einstein\'s contraction of the tensor product......Page all_16951_to_0225.cpc0066.djvu
3.6 Matrix representation of tensors......Page all_16951_to_0225.cpc0068.djvu
3.7 Exercises......Page all_16951_to_0225.cpc0073.djvu
4.2 Change of basis in a third-order tensor product space......Page all_16951_to_0225.cpc0076.djvu
4.3 Matrix representation of a change-of-basis in tensor spaces......Page all_16951_to_0225.cpc0078.djvu
4.4 General criteria for tensor character......Page all_16951_to_0225.cpc0080.djvu
4.5 Extension to homogeneous tensors......Page all_16951_to_0225.cpc0083.djvu
4.6 Matrix operation rules for tensor expressions......Page all_16951_to_0225.cpc0085.djvu
4.8 Main isotropic tensors......Page all_16951_to_0225.cpc0091.djvu
4.9 Exercises......Page all_16951_to_0225.cpc0117.djvu
5.2 Main theorem on tensor contraction......Page all_16951_to_0225.cpc0121.djvu
5.3 The contracted tensor product and tensor homomorphisms......Page all_16951_to_0225.cpc0123.djvu
5.4 Tensor product applications......Page all_16951_to_0225.cpc0129.djvu
5.5 Criteria for tensor character based on contraction......Page all_16951_to_0225.cpc0132.djvu
5.6 The contracted tensor product in the reverse sense: The quotient law......Page all_16951_to_0225.cpc0134.djvu
5.7 Matrix representation of permutation homomorphisms......Page all_16951_to_0225.cpc0137.djvu
5.8 Matrices associated with simply contraction homomorphisms......Page all_16951_to_0225.cpc0151.djvu
5.9 Matrices associated with doubly contracted homomorphisms......Page all_16951_to_0225.cpc0154.djvu
5.10 Eigentensors......Page all_16951_to_0225.cpc0169.djvu
5.11 Generalized multilinear mappings......Page all_16951_to_0225.cpc0175.djvu
5.12 Exercises......Page all_16951_to_0225.cpc0193.djvu
Part II - Special Tensors......Page all_16951_to_0225.cpc0197.djvu
6.2 Symmetric systems of scalar componsnts......Page all_16951_to_0225.cpc0198.djvu
6.3 Strict components of a symmetric system......Page all_16951_to_0225.cpc0200.djvu
6.4 Tensors with symmetries: Tensors with branched symmetry, symmetric tensors......Page all_16951_to_0225.cpc0202.djvu
6.5 Symmetric tensors under the tensor algebra perspective......Page all_16951_to_0225.cpc0215.djvu
6.6 Symmetric tensor algebras: The symmetric tensor product......Page all_16951_to_0225.cpc0221.djvu
6.7 Illustrative examples......Page all_16951_to_0225.cpc0223.djvu
6.8 Exercises......Page all_16951_to_0450.cpc0004.djvu
7.2 Anti-symmetric systems of scalar components......Page all_16951_to_0450.cpc0008.djvu
7.3 Strict components of an anti-symmetric system and with respect to an index subset......Page all_16951_to_0450.cpc0011.djvu
7.4 Tensors with anti-symmetries: Tensors with branched anti-symmetry; anti-symmetric tensors......Page all_16951_to_0450.cpc0013.djvu
7.5 Anti-symmetric tensors from the tensor algebra perspective......Page all_16951_to_0450.cpc0029.djvu
7.6 Anti-symmetric tensor algebras: The anti-symmetric tensor product......Page all_16951_to_0450.cpc0035.djvu
7.7 Illustrative examples......Page all_16951_to_0450.cpc0036.djvu
7.8 Exercises......Page all_16951_to_0450.cpc0048.djvu
8.2 Previous concepts of modular tensor establishment......Page all_16951_to_0450.cpc0051.djvu
8.3 Axiomatic properties for the modular tensor concept......Page all_16951_to_0450.cpc0052.djvu
8.4 Modular tensor characteristics......Page all_16951_to_0450.cpc0053.djvu
8.5 Remarks on modular tensor operations: Consequences......Page all_16951_to_0450.cpc0054.djvu
8.6 Modular symmetry and anti-symmetry......Page all_16951_to_0450.cpc0062.djvu
8.7 Main modular tensors......Page all_16951_to_0450.cpc0073.djvu
8.8 Exercises......Page all_16951_to_0450.cpc0092.djvu
Part III - Exterior Algebras......Page all_16951_to_0450.cpc0095.djvu
9.1 Introduction and definitions......Page all_16951_to_0450.cpc0096.djvu
9.2 Exterior product of r vectors: Decomposable multivectors......Page all_16951_to_0450.cpc0100.djvu
9.3 Axiomatic properties of tensor operations in exterior algebras......Page all_16951_to_0450.cpc0105.djvu
9.4 Dual exterior algebras over V*ⁿ(K) spaces......Page all_16951_to_0450.cpc0112.djvu
9.5 The change-of-basis in exterior algebras......Page all_16951_to_0450.cpc0118.djvu
9.6 Complements of contramodular and comodular scalars......Page all_16951_to_0450.cpc0122.djvu
9.8 Scalar mappings: Exterior contractions......Page all_16951_to_0450.cpc0123.djvu
9.9 Exterior vector mappings: Exterior homomorphisms......Page all_16951_to_0450.cpc0126.djvu
9.10 Exercises......Page all_16951_to_0450.cpc0164.djvu
10.1 Introduction......Page all_16951_to_0450.cpc0167.djvu
10.2 Decomposable mixed exterior vectors: mixed exterior product of p vectors Vi and q vectors Vj......Page all_16951_to_0450.cpc0174.djvu
10.3 Mixed exterior algebras: Terminology......Page all_16951_to_0450.cpc0177.djvu
10.4 Exterior product of mixed exterior vectors......Page all_16951_to_0450.cpc0179.djvu
10.5 Anti-commutativity of the ^ mixed exterior product......Page all_16951_to_0450.cpc0183.djvu
10.6 Change of basis in mixed exterior algebras......Page all_16951_to_0450.cpc0184.djvu
10.7 Exercises......Page all_16951_to_0450.cpc0189.djvu
Part IV - Tensors over Linear Spaces with Inner Product......Page all_16951_to_0450.cpc0191.djvu
11.2 Initial concepts......Page all_16951_to_0450.cpc0192.djvu
11.3 Tensor character of the inner vector\'s connection in a PSEⁿ(R) space......Page all_16951_to_0450.cpc0195.djvu
11.4 Different types of the fundamental connection tensor......Page all_16951_to_0450.cpc0197.djvu
11.5 Tensor product of vectors in Eⁿ(R) (or in PSEⁿ(R))......Page all_16951_to_0450.cpc0200.djvu
11.6 Equivalent associated tensors: Vertical dispacements of indices. Generalization......Page all_16951_to_0450.cpc0201.djvu
11.7 Changing bases in Eⁿ(R): Euclidean tensor character criteria......Page all_16951_to_0450.cpc0206.djvu
11.8 Symmetry and anti-symmetry in Euclidean tensors......Page all_16951_to_0450.cpc0209.djvu
11.9 Cartesian tensors......Page all_16951_to_0450.cpc0212.djvu
11.10 Euclidean tensor algebra or pseudo-Euclidean tensor algebra......Page all_16951_to_0674.cpc0005.djvu
11.11 Euclidean tensor metrics: The (r ☺ 1 Eⁿ) (R) space as Euclidean space......Page all_16951_to_0674.cpc0036.djvu
11.12 Exercises......Page all_16951_to_0674.cpc0058.djvu
12.2 Diverse cases of linear space connections......Page all_16951_to_0674.cpc0064.djvu
12.3 Tensor character of √|G|......Page all_16951_to_0674.cpc0065.djvu
12.5 Tensor character of the orientation tensor......Page all_16951_to_0674.cpc0067.djvu
12.6 Orientation tensors as associated Euclidean tensors......Page all_16951_to_0674.cpc0068.djvu
12.7 Dual or polar tensors over Eⁿ(R) Euclidean spaces......Page all_16951_to_0674.cpc0069.djvu
12.8 Exercises......Page all_16951_to_0674.cpc0078.djvu
13.2 Euclidean exterior algebra of order r = 2......Page all_16951_to_0674.cpc0081.djvu
13.3 Euclidean exterior algebra of order r (2 < r < n)......Page all_16951_to_0674.cpc0084.djvu
13.5 The orientation tensor in exterior bases......Page all_16951_to_0674.cpc0087.djvu
13.6 Dual or polar tensors in exterior bases......Page all_16951_to_0674.cpc0088.djvu
13.7 The cross product as a polar tensor in generalized Cartesian coordinate frames......Page all_16951_to_0674.cpc0090.djvu
13.8 √|G| geometric interpretation in generalized Cartesian coordinate frames......Page all_16951_to_0674.cpc0091.djvu
13.9 Illustrative examples......Page all_16951_to_0674.cpc0092.djvu
13.10 Exercises......Page all_16951_to_0674.cpc0128.djvu
Part V - Classic Tensors in Geometry and Mechanics......Page all_16951_to_0674.cpc0130.djvu
14.1 Introduction and motivation......Page all_16951_to_0674.cpc0131.djvu
14.2 Euclidean tensors in Eⁿ(R)......Page all_16951_to_0674.cpc0132.djvu
14.3 Affine geometric tensors. Homographies......Page all_16951_to_0674.cpc0147.djvu
14.4 Tensors in Physics and Mechanics......Page all_16951_to_0674.cpc0176.djvu
14.5 Exercises......Page all_16951_to_0674.cpc0205.djvu
References......Page all_16951_to_0674.cpc0209.djvu
Index......Page all_16951_to_0674.cpc0213.djvu
Universitext......Page all_16951_to_0674.cpc0221.djvu