Two-point tensor

Two-point tensors, or double vectors, are tensor-like quantities which transform as 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 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, A_jM.

Continuum mechanics:While a usual tensor can be viewed as a transformation of a vector in a given co-ordinate space to another vecto in the same co-ordinate space a two point tensor can transform a vector from one co-ordinate system to another. So just as an ordinary tensor $Q{_{p}}{_{q}}(e{_{p}}\otimes e{_{q}})$ transforms a vector u to a vector v such that v=Qu where v and u are measured in the same co-ordinate system denoted by 'e'. A two point G will be written as $G{_{p}}{_{q}}(e{_{p}}\otimes E{_{q}})$ will transform a vector U in the 'E' system to a vector 'v' in the 'e' system as ${v=GU}$

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
$v'{_{p}}=Q{_{p}}{_{q}}v{_{q}}$
For tensors suppose we then have
$T{_{p}}{_{q}}(e{_{p}}\otimes e{_{q}})$ ,A tensor in the system $e{_{i}}$
In another system let the same tensor be given by $T'{_{p}}{_{q}}(e'{_{p}}\otimes e'{_{q}})$
We can of course say
T'_ij=Q_ipQ_jrT_pr
Then T'=QTQ^T is the routine tensor transformation

But a two point tensor between these systems is just$F{_{p}}{_{q}}(e'{_{p}}\otimes e{_{q}})$ which transforms as
F'=QF

The most mundane example of a two point tensor::

The most mundane example of a two point tensor is the harmless looking Transformation tensor .The Q in the above discussion Note that v'_p=Q_pqu_q
Now writing out in full u=u_qe_q
and also v'=v'_pe_p
This then requires Q to be of the form $Q{_{p}}{_{q}}(e'{_{p}}\otimes e{_{q}})$

Clearly by definition of tensor product $(e'{_{p}}\otimes e{_{q}})e{_{q}}=(e{_{q}}.e{_{q}})e'{_{p}}=e'{_{p}}$...........................(1)
So we can write that:
$u{_{p}}e{_{p}}=(Q{_{p}}{_{q}}(e'{_{p}}\otimes e{_{q}}))(v{_{q}}e{_{q}})$
Thus $u{_{p}}e{_{p}}=Q{_{p}}{_{q}}v{_{q}}(e'{_{p}}\otimes e{_{q}})e{_{q}}$ This on incorporating (1) we have

$u{_{p}}e{_{p}}=Q{_{p}}{_{q}}v{_{q}}e{_{p}}$ as consistency requires

--Bilwaj (talk) 12:33, 28 November 2007 (UTC) Template:Engr-stub