Tensors

In General > s.a. tensor fields [tensor densities, calculus]; types of fiber bundles [tensor bundles].
* History: Tensors were first fully described in the 1890s by Gregorio Ricci-Curbastro, with the help of his student Tullio Levi-Civita, and they were given their name in 1898 by Woldemar Voigt, a German crystallographer, who was studying stresses and strains in non-rigid bodies.
$ Def 1: (Cartan's point of view) A (p, q)-tensor over a vector space V is a multilinear map from p copies of V* and q copies of V to a field (in practice, \(\mathbb R\) or \(\mathbb C\)),

T: V* × V* × ... V* × V × V × ... V → \(\mathbb R\).

$ Def 2: (Transformation point of view) A (p, q)-tensor over a vector space V is an object which, under a change of basis for V represented by the matrix A, transforms under

T' ^{a.... b}_{c... d} = A^a_m ... A^b_n T ^{m... n}_{p... q} \((A^{-1})^p{}_c \ldots (A^{-1})^q{}_d\) .

* Special cases: If V is real n-dimensional, tensors of type (p, q) on V carry a representation of GL(n, \(\mathbb R\)).
* Symmetry properties: A p-th order covariant (for example) tensor T has the symmetry (or antisymmetry) defined by π ∈ S_p if πT = T (or πT = (−1)^sign(π) T), where the action of π is defined by πT(v₁, ..., v_p):= T(v_π(1), ..., v_π(p)).
@ Student guides: Fleisch 11; Battaglia & George AJP(13)jul [undergraduate].
@ And physics: Joshi 95; Jeevanjee 11 [r PT(12)apr].
@ Other references: Olive & Auffray MMCS-a1301 [symmetry classes for even-order tensors]; Kissinger a1308 [abstract indices, categorical treatment].
> Online resources: see Marcus Hanke's page; MathWorld page.

Tensor Algebra and Operations > s.a. Algebraic Geometry [decomposition of tensors]; computation [including symbolic manipulation].
* Idea: The set of tensors of type (p, q) is a linear space, while the set of all tensors forms an algebra with the operations of addition and tensor product; Additional operations defined on it are contraction, trace, ...
* Identity: For any two rank-2 antisymmetric tensors A_ab and B_ab in 4D, A B – *A*B = \(1\over2\) delta A B.
$ Symmetrization: Given a p-th order tensor T, the action of the symmetrization operator A on it is

S T:= \(1\over p!\)∑_{π ∈ Sp} πT .

$ Antisymmetrization: Given a p-th order tensor T, the action of the antisymmetrization operator A on it is

A T:= \(1\over p!\)∑_{π ∈ Sp} (signature of π) πT ,

or, in components, (AT)_{i1, ..., ip}:= \(1\over p!\)ε_{i1, ..., ip}^{k₁, ..., k_p} T_{k1, ..., kp}.
* With a metric:
@ References: Edgar & Höglund JMP(02)gq/01 [generalized Lovelock identity]; Portugal & Svaiter mp/01, Manssur & Portugal IJMPC(02)mp/01 [symbolic manipulation].

Tensor Product between Tensors > s.a. metric tensor.
$ Def: Given, for example, the two tensors u ∈ ⊗^q T_xX and θ ∈ ⊗^p T*_x X, their tensor product is the tensor T = u ⊗ θ ∈ {⊗^q T_x X ⊗^p T*_x X}, defined by

T(ω₁, ..., ω_q, v₁, ..., v_p):= u(ω₁, ..., ω_q) θ(v₁, ..., v_p) ,

or, using abstract index notation, T ^{a... b}_{c... d}:= u^{a... b} θ_{c... d} .

Tensor Product between Vector Spaces
$ Def: V ⊗ W:= {f : V* × W* → \(\mathbb R\) (\(\mathbb C\)), f bilinear}, with (af + bg) (ξ, η):= a f(ξ, η) + b g(ξ, η).
* In practice: If {v_i} is a basis for V and {w_m} one for W, V ⊗ W:= {f = ∑_im c_im v_i ⊗ w_m | c_im ∈ \(\mathbb R\) (\(\mathbb C\))}.
* With Hilbert space structure: If f = ∑_im c_im v_i ⊗ w_m and g = ∑_im d_im v_i ⊗ w_m, then \(\langle\)f, g\(\rangle\):= ∑_ijnm c_im d_jn \(\langle v_i, v_j \rangle_V\, \langle w_m, w_n \rangle_W\) .
@ Between Banach spaces: Grothendieck BSMSP(56) [tensor norms].

Generalizations > s.a. Holors; quantum states [tensor network factorization]; tensor fields.
@ References: Fernández et al AACA(01)mp/02 ["extensors"]; Gaete & Wotzasek PLB(06) [negative rank?]; Christandl & Zuiddam CC(18)-a1606 [tensor surgery].

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