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 π*T*
= *T* (or π*T* = (−1)^{sign(π) }*T*),
where the action of π is defined by π*T*(*v*_{1},
..., *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} π*T* .

$ __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) π*T* ,

or, in components, (*AT*)_{i1,
..., ip}:=
\(1\over p!\)*ε*_{i1,
..., ip}^{k1,
..., kp} *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_{x}*X* 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*(*ω*_{1},
..., *ω*_{q},
*v*_{1},
..., *v*_{p}):=
*u*(*ω*_{1},
..., *ω*_{q})
*θ*(*v*_{1},
..., *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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