A tensor is a particular kind of mathematical quantity associated with a geometry. In particular, they are a class of representations for the group of congruencies for that geometry, meaning that under a change of coordinate systems their components transform in a linear way. Scalars and vectors are both examples of tensors. Einstein resurrected tensors from relative obscurity in order to express general relativity in a coordinate independent way.
Mathematically, given any two vector spaces V,W over some common field F, we define their tensor product V Ä W to be the space unique up to isomorphism such that any bilinear mapping V x W -> F can be expressed as the composition of a linear map V Ä W -> F and a particular bilinear map V x W -> V Ä W. This needs to be elaborated on, but basically what it means is that for {ei} and {fj} bases for V and W, one can define a basis {ei Ä fj} for V Ä W in a natural way.
So the dimension of a tensor product is the product of the dimensions of the spaces. Note that F Ä V is naturally isomorphic to V, and V* Ä W (where V* denotes the dual space of W, whose basis is usually denoted {ei} is isomorphic in a natural way to the space of linear transformations (ie matrices) from V to W. An inner product V x V -> F corresponds in a natural way to a tensor in V Ä V, called the associated metric and usually denoted g.
In differential geometry and physics we usually deal with tensor fields. At any given point the value is taken to be in the space V Ä V Ä ... Ä V Ä V* Ä V* Ä ... Ä V*, where V is the tangent space at a particular point on our manifold and V* is the cotangent space. If there are m copies of V and n copies of V* in our product, the tensor is said to be of contravariant rank m and covariant rank n. The tensors of rank zero are just the scalars F, those of contravariant rank 1 our tangent vectors V, and those of covariant rank 1 our one-forms V* (for this reason the last two spaces are often called the contravariant and covariant vectors).
For any given coordinate system we have a basis {ei} for the tangent space V (note that this may vary from point-to-point if the manifold is not linear), and a corresponding basis {ei} for the cotangent space V* (see dual space someday). The difference between the raised and lowered indices is there to remind us of the way the components transform, and by convention (Einstein notation) it is understood that when a quantity appears both raised and lowered on the same side of an equation, we are summing over all its possible values.
For example purposes, then, take a tensor A in the space V Ä V Ä V*. The components relative to our component system can be written
A = Aijk (ei x ej x ek)
and in physics we often use the expression Aijk to represent the tensor, just as vectors are usually treated in terms of their components. This can be visualized as an n x n x n array of numbers. In a different coordinate system, say giving us a basis of {ei'}, the components will be different. If xi'i is our transformation matrix (note ir is not a tensor, since it represents a change of basis rather than of the space), then our components vary per
Ai'j'k' = xi'i xjj' xk'k Aijk
In older texts this transformation rule often serves as the definition of a tensor.
This page seems to be really good. Perhaps it should be combined with the main page?