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}*
... (

* __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].

* __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 a1606 [tensor surgery].

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jun 2016