Tensors

In General > s.a. tensor fields [tensor calculus]; types of fiber bundles [tensor bundles].
$ 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, R or C),

T: V* × V* × ... V* × V × V × ... V → 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 ... (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, R).
* Symmetry properties: A p-th order covariant (for example) tensor T has the symmetry (or antisymmetry) defined by S_p if T = T (or T = (–1)^sign(pi) T), where the action of is defined by T(v₁, ..., v_p):= T(v_pi(1), ..., v_pi(p)).
@ Introductory text: Joshi 95 [and physics].

Tensor Algebra and Operations
* 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, ...
$ Symmetrization: Given a p-th order tensor T, the action of the symmetrization operator A on it is

S T:= (1/p!) _{pi in S_p} T .

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

A T:= (1/p!) _{pi in S_p} (signature of ) T ,

or, in components, (AT)_{i_1, ..., i_p}:= (p!)^–1 _{i_1, ..., i_p}^{k_1, ..., k_p} T_{k_1, ..., k_p}.
* With a metric:
@ References: Edgar & Höglund JMP(02)gq/01 [generalized Lovelock identity]; Portugal & Svaiter mp/01, Manssur & Portugal mp/01 [symbolic manipulation].

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

T(₁, ..., _q, v₁, ..., v_p):= u(₁, ..., _q) (v₁, ..., v_p) ,

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

Tensor Product between Vector Spaces
$ Def: V W:= {f : V* × W* → R (C), f bilinear}, with (af + bg) (,):= a f(,) + b g(,).
* 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 R (C)}.
* With Hilbert space structure: If f = _im c_im v_i w_m and g = _im d_im v_i w_m, then f, g:= _ijnm c_im d_jn v_i,v_jV w_m,w_nW .
@ Between Banach spaces: Grothendieck BSMSP(56) [tensor norms].

Generalizations > see Holors; tensor fields.
@ References: Fernández et al AACA(01)mp/02 ["extensors"]; Gaete & Wotzasek PLB(06) [negative rank?].

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