Note: This is a fairly abstract mathematical approach to tensors. If you are baffled by this article, try reading the main tensor article and the classical treatment first.

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 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.

Definition

Let V and W be two real vector spaces. Their tensor product is a real vector space

together with a bilinear map

If {e_i} and {f_j} are bases for V and W, the set

is a basis for this tensor product, the dimension of which is given by the product of the dimensions of V and W. (Just to avoid confusion, note that here the same symbol has been used with two different--albeit related--senses, one for vector spaces, and one for individual vectors.) This tensor product can be generalized 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:

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 rank m and covariant rank n. The tensors of rank zero are just the scalars R, 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).

Note that 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 V × V → R corresponds in a natural way to a (0,2) tensor in

called the associated metric and usually denoted g.

In differential geometry, physics and engineering, we usually deal with tensor fields on differentiable manifolds. (The term "tensor" is sometimes used as a shorthand for "tensor field".) For instance, the curvature tensor is discussed in differential geometry and the stress-energy tensor is important in physics and engineering. Both of these are related by Einstein's theory of general relativity. In engineering, the underlying manifold will often be Euclidean 3-space. A tensor field assigns to any given point of the manifold a tensor in the space

where V is the tangent space at that point and V* is the cotangent space. See also tangent bundle and cotangent bundle.

For any given coordinate system we have a basis {e_i} for the tangent space V (note that 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 {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

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