integrate by parts double derivative, possible bug?

Question

Hi, I think I may have found a bug using integrate by parts in double derivative. The code is the following

{\mu,\nu,\rho}::Indices(position=free). 
x::Coordinate; 
\partial{#}::PartialDerivative; 
\delta{#}::Accent;
A_{\mu}::Depends(x,\partial{#},\delta{#});
S:= -1/4 \int{ \alpha A_{\mu} \partial_{\rho}{\partial^{\rho}{A^{\mu}}} }{x};
integrate_by_parts(_,$A^{\mu}$);

I think the following output is incorrect because it should be with (-1/4):

$$(1/4)\alpha\int{\partial_{\rho}{}^{\rho}{A_{\mu}} A^{\mu}}dx. $$

Someone knows what's going on?

thank you in advance,

Victor

ps: I have edited the previous post caused there were no bug between unwrap after varying.

Answer 1

Thanks, this is indeed a bug, caused by the incorrect handling of multiple indices on a partial derivative. As a workaround you can declare \partial{#}::Derivative to prevent multiple indices on the derivative, and then do the integrate_by_parts twice (you then also need to declare the \mu,\nu,\rho indices to be position=fixed). Will put this on the todo list.

Comments

Thanks to this answer I when back to the manual of Indices and found a very clear explanation of the different position's values. Great job!!!