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In mathematics, the modern component-free approach to the theory of tensors views tensors initially as abstract objects, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference 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 or intermediate level treatments first.
Definition via tensor products of vector spaces
Given a finite set { V1, ... , Vn } of vector spaces over a common field F. One may form their tensor product V1 ? ... ? Vn. An element of this tensor product is referred to as a tensor (but this is not the notion of tensor discussed in this article).
A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form:
where V* is the dual space of V.
If there are m copies of V and n copies of V* in our product, the tensor is said to be of type (m, n) and of contravariant order m and covariant order n and total order m+n. The tensors of order zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are 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 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
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
Another, alternate notation for this space is in terms of linear maps from a vector space V to a vector space W. Let
denote the space of all linear maps from V to W. Thus, for example, the dual space (the space of linear functionals) may be written as
The set of (m,n)-tensors can then be written as
In the formula above,the roles of V and V* are reversed. In particular, one has
and
and
The notation
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 fields on smooth manifolds. 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
For any given coordinate system we have a basis {ei} for the tangent space V (this may vary from point-to-point if the manifold is not linear), and a corresponding dual basis {ei} for the cotangent space V* (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
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
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 {ei'}, the components will be different. If (xi'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 (yii') is its inverse, then our components vary per
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.
References
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