Tensor (intrinsic Definition)

In Differential Geometry an intrinsic geometric statement may be described by a Tensor Field on a Manifold , and then doesn't need to make references to coordinates at all. The same is true in General Relativity , of tensor fields describing a Physical Property . The component-free approach is also used heavily in Abstract Algebra and Homological Algebra , where tensors arise naturally.

''Note: This article, which is fairly abstract, requires an understanding of the Tensor Product of Vector Spaces without chosen Bases . The notion of a tensor product generalizes to vector spaces without chosen bases, and even further, to Modules . If you find this article difficult, try reading the main Tensor article and the Classical Treatment first.''

DEFINITION: TENSOR PRODUCT OF VECTOR SPACES

Let ''V'' and ''W'' be two Vector Spaces over a common Field F. Their Tensor Product

:

V \otimes_F W

is a vector space over the same field F together with a Bilinear Map

:

\otimes:  V 	imes W 
arr V \otimes W

which is universal (i.e., the smallest possible without throwing away information) in the following sense:

for every vector space ''X'' over the field F and every F-bilinear map

:

Q:  V 	imes W 
arr X \,

there is a unique F-linear map

:

Q': V \otimes_F W 
arr X

such that

:

Q (v,w)  = Q'(v\otimes w) \ \ orall v \in V, w \in W.

It is easy to see that a vector space

V \otimes_F W

is unique up to isomorphism if it exists, and we write ''the'' instead of ''a'' tensor product. All its properties, except its existence, follow from the abstract definition, although some properties are more easily understood from an explicit model. An explicit construction is easy to give using a Bases {v_i} and {'''w'''_j} respectively. The tensor product

V \otimes W

can be constructed as the vector space spanned by a Basis

:

\{ \mathbf{v}_i  \otimes \mathbf{w}_j \}

where in the basis, the symbol

\otimes

is alternatively seen as a formal symbol for forming a pair, and the value of the bilinear map

\otimes

on the basis vectors. The extention of

\otimes

to all of

V 	imes W

is done in the unique way compatible with bilinearity.

If ''V'' and ''W'' are both finite dimensional then the Dimension of

V \otimes W

is the product of the dimensions of ''V'' and ''W''. This tensor product can be repeated to apply to more than just two vector spaces.

A tensor on the vector space ''V'' is then defined to be an element of (i.e., a vector in) the following vector space:

\otimes ... \otimes V^---

is the Dual Space of ''V''.

in our product, the tensor is said to be of type (''m'', ''n'') and of contravariant rank ''m'' and covariant rank ''n'' and total rank ''m+n''. The tensors of rank zero are just the scalars (elements of the field F), those of contravariant rank 1 the vectors in V, and those of covariant rank 1 the one-forms in V--- (for this reason the last two spaces are often called the contravariant and covariant vectors).

The (1,1) tensors

are isomorphic in a natural way to the space of linear transformations (i.e., Matrices ) from V to V. An Inner Product of a real vector space V; V × V → R corresponds in a natural way to a (0,2) tensor in

\otimes V^---

called the associated Metric and usually denoted ''g''.

ALTERNATE NOTATION

Rather than writing out the full tensor product to denote the space of tensors of type (m,n), the literature often uses the abbreviation

... \otimes V^---

L(V,W)\

denote the space of all linear maps from V to W. Thus, for example, the dual space (the space of 1-form s) may be written as

\approx L(V,\mathbb{R})

T^m_n(V) \approx

\otimes ... \otimes V^---\otimes V \otimes ... \otimes V, \mathbb{R})
,...,V^---,V,...,V,\mathbb{R})

are reversed.

,\mathbb{R}) \approx V

T^1_1(V) \approx L(V,V)

The notation

:

GL(V,W)\

is often used to denote the space of invertible linear transformations from V to W; however there is no analogous notation for tensor spaces.

TENSOR FIELDS

''See main article Tensor Field ''

Differential Geometry , Physics and Engineering must often deal with Tensor Field s on Smooth Manifold s. The term ''tensor'' is in fact sometimes used as a shorthand for ''tensor field''. A tensor field expresses the concept of a tensor that varies from point to point.

BASIS

(see Dual Space ). The difference between the raised and lowered indices is there to remind us of the way the components transform.

For example purposes, then, take a tensor A in the space

The components relative to our coordinate system can be written

:

\mathbf{A} = A^{ij} {}_k (\mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{e}^k)

Here we used the Einstein Notation , a convention useful when dealing with coordinate equations: when an index variable appears both raised and lowered on the same side of an equation, we are summing over all its possible values. In physics we often use the expression

:

A^{ij} {}_k\

to represent the tensor, just as Vectors are usually treated in terms of their components. This can be visualized as an ''n'' × ''n'' × ''n'' array of numbers. In a different coordinate system, say given to us as a basis {e_i'}, the components will be different. If (x^i'_i) is our transformation matrix (note it is not a tensor, since it represents a change of basis rather than a geometrical entity) and if (yⁱ_i') is its inverse, then our components vary per

:

A^{i'j'}\! {}_{k'} = x^{i'}\! {}_i \, x^{j'}\! {}_j \, y^k\! {}_{k'} \, A^{ij} {}_k

In older texts this transformation rule often serves as the definition of a tensor. Formally, this means that tensors were introduced as specific Representations of the Group of all changes of coordinate systems.

/Old Talk - still has some stuff that should likely be merged in

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Tensor (intrinsic Definition)