Two-point tensors, or double vectors, are tensor-like quantities which transform as Euclidean vectors with respect to each of their indices. They are used in continuum mechanics to transform between reference ("material") and present ("configuration") coordinates. 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 two-point tensor will have one capital and one lower-case index; for example, "AjM".
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 two-point tensor transforms vectors from one coordinate system to another. That is, a conventional tensor,
 formula_1,
actively transforms a vector u to a vector v such that 
formula_2
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 two-point tensor, G will be written as
 formula_3
and will transform a vector, U, in E system to a vector, v, in the e system as 
formula_4.
The transformation law for two-point tensor.
Suppose we have two coordinate systems one primed and another unprimed and a vectors' components transform between them as 
formula_5.
For tensors suppose we then have
formula_6.
A tensor in the system formula_7. In another system, let the same tensor be given by
 formula_8.
We can say
formula_9.
Then 
formula_10
is the routine tensor transformation. But a two-point tensor between these systems is just
 formula_11
which transforms as
 formula_12.
Simple example.
The most mundane example of a two-point tensor is the transformation tensor, the "Q" in the above discussion. Note that
 formula_13.
Now, writing out in full,
formula_14
and also
formula_15.
This then requires "Q" to be of the form
 formula_16.
By definition of tensor product,
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So we can write
 formula_17
Thus
 formula_18
Incorporating (1), we have
formula_19.
References.
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