Two-point tensor

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Two-point tensors, or double vectors, are

.

As with many applications of tensors,

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, A_jM.

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,

${\displaystyle \mathbf {Q} =Q_{pq}(\mathbf {e} _{p}\otimes \mathbf {e} _{q})}$,

a vector u to a vector v such that

${\displaystyle \mathbf {v} =\mathbf {Q} \mathbf {u} }$

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

${\displaystyle \mathbf {G} =G_{pq}(\mathbf {e} _{p}\otimes \mathbf {E} _{q})}$

and will transform a vector, U, in E system to a vector, v, in the e system as

${\displaystyle \mathbf {v} =\mathbf {GU} }$.

Suppose we have two coordinate systems one primed and another unprimed and a vectors' components transform between them as

${\displaystyle v'_{p}=Q_{pq}v_{q}}$.

For tensors suppose we then have

${\displaystyle T_{pq}(e_{p}\otimes e_{q})}$.

A tensor in the system ${\displaystyle e_{i}}$. In another system, let the same tensor be given by

${\displaystyle T'_{pq}(e'_{p}\otimes e'_{q})}$.

We can say

${\displaystyle T'_{ij}=Q_{ip}Q_{jr}T_{pr}}$.

Then

${\displaystyle T'=QTQ^{\mathsf {T}}}$

is the routine tensor transformation. But a two-point tensor between these systems is just

${\displaystyle F_{pq}(e'_{p}\otimes e_{q})}$

which transforms as

${\displaystyle F'=QF}$.

The most mundane example of a two-point tensor is the transformation tensor, the Q in the above discussion. Note that

${\displaystyle v'_{p}=Q_{pq}u_{q}}$.

Now, writing out in full,

${\displaystyle u=u_{q}e_{q}}$

and also

${\displaystyle v=v'_{p}e'_{p}}$.

This then requires Q to be of the form

${\displaystyle Q_{pq}(e'_{p}\otimes e_{q})}$.

By definition of tensor product,

${\displaystyle (e'_{p}\otimes e_{q})e_{q}=(e_{q}.e_{q})e'_{p}=3e'_{p}}$

1

So we can write

${\displaystyle u_{p}e_{p}=(Q_{pq}(e'_{p}\otimes e_{q}))(v_{q}e_{q})}$

Thus

${\displaystyle u_{p}e_{p}=Q_{pq}v_{q}(e'_{p}\otimes e_{q})e_{q}}$

Incorporating (1), we have

${\displaystyle u_{p}e_{p}=Q_{pq}v_{q}e_{p}}$.

• Mixed tensor
• Covariance and contravariance of vectors

• Mathematical foundations of elasticity By Jerrold E. Marsden, Thomas J. R. Hughes
• Two-point Tensors at iMechanica