my description of tensors

How's this for a description of what tensors are all about:

A tensor can be a scalar, vector, matrix, or higher dimensional array (it is
a generalisation of arrays).

Tensors can be used to transform from one coordinate system to another.

A tensor in the original coordinate system is called a covariant tensor.
A tensor in the transformed coordinate system (sometimes called the 'dual
space') is called a contravariant tensor.

A covariant (original) tensor can be multiplied by a 'metric tensor' to
produce a contravariant (transformed) tensor.

If we use the Lorentz tensor as the metric tensor then we have the
relativistic transformation of special relativity.
(i.e. it is the metric/Lorentz tensor that transforms the original tensor)

....so a 'mixed tensor' would be a tensor with some indices of the original
coordinate system and some indices of the transformed coordinate system.


How's that? Have I got it yet?