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Two-point tensors, or double vectors, are tensor-like 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. 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.
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,
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 two-point tensor, G will be written as
and will transform a vector, U, in E system to a vector, v, in the e system as
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
is the routine tensor transformation. But a two-point tensor between these systems is just
which transforms as
The most mundane example of a two-point tensor is the transformation tensor, the Q in the above discussion. Note that
Now, writing out in full,
This then requires Q to be of the form
By definition of tensor product,
So we can write
Incorporating (1), we have
In the equation following (1) there are four q's !?