[GiNaC-list] Differentiation of a function with respect to a tensor

[Stephen Montgomery-Smith]

Bernardo Rocha wrote:
> Hi everyone,
>
> I've recently discovered GiNaC and I'm really excited about its
> capabilities. There is one thing that I would like to know if it is able
> to do that I haven't found in the tutorial.pdf or in any other place
> that I've searched.
>
> I would like to know if, given a function \Psi=\Psi(E), like the strain
> energy function for the St. Venant-Kirchhoff material
>
> \Psi(E) = 0.5 * \lambda * (tr E)^2 + \mu E:E
>
> is it possible to differentiate it with respect to E, that is i would
> like to compute \frac{\partial \Psi}{\partial E}. If this is possible,
> could someone please send some examples or maybe point to which classes
> should I use to do that?
>
> That's all for now. Many thanks in advance.
>
> Best regards,
> Bernardo M. R.
>
>

Couldn't you do it this way?  Write \Psi(E) as an expression involving 
the variables e11,e12,e13,...,e33 which are the entries of E.  Then 
compute the partial derivatives \frac{\partial \Psi}{\partial eij} for 
1<=i,j<=3.  (Presumably you suppose that E is symmetric so only six 
partial derivatives need to be computed, but even if it is not 
necessarily symmetric you still only need 9 partial derivatives.)  Just 
store this as something like:

expr dPsi_dE[3][3]

or

vector<expr> dPsi_dE

or something similar.