Twopoint tensor
Twopoint tensors, or double vectors, are tensorlike quantities which transform as euclidean vectors with respect to each of their indices and are used in continuum mechanics to transform between reference ("material") and present ("configuration") coordinates.[1] Examples include the deformation gradient and the first Piola–Kirchhoff stress tensor.
As with many applications of tensors, Einstein summation notation is frequently used. To clarify this notation, capital indices are often used to indicate reference coordinates and lowercase for present coordinates. Thus, a twopoint tensor will have one capital and one lowercase index; for example, A_{jM}.
Continuum mechanics
A conventional tensor can be viewed as a transformation of vectors in one coordinate system to other vectors in the same coordinate system. In contrast, a twopoint tensor transforms vectors from one coordinate system to another. That is, a conventional tensor,
 ,
actively transforms a vector u to a vector v such that
where v and u are measured in the same space and their coordinates representation is with respect to the same basis (denoted by the "e").
In contrast, a twopoint tensor, G will be written as
and will transform a vector, U, in E system to a vector, v, in the e system as
 .
The transformation law for twopoint tensor
Suppose we have two coordinate systems one primed and another unprimed and a vectors' components transform between them as
 .
For tensors suppose we then have
 .
A tensor in the system . In another system, let the same tensor be given by
 .
We can say
 .
Then
is the routine tensor transformation. But a twopoint tensor between these systems is just
which transforms as
 .
The most mundane example of a twopoint tensor
The most mundane example of a twopoint tensor is the transformation tensor, the Q in the above discussion. Note that
 .
Now, writing out in full,
and also
 .
This then requires Q to be of the form
 .
By definition of tensor product,

(1)
So we can write
Thus
Incorporating (1), we have
 .
In the equation following (1) there are four q's !?
References
 Humphrey, Jay D. Cardiovascular solid mechanics: cells, tissues, and organs. Springer Verlag, 2002.