How do you integrate eqns with indices?

[JuanC97]

Hello, I just want to clarify some things with a simple exercise: I have the equation $\frac{\partial^2 f}{\partial A^\mu \,\partial A^\nu} = 0$ and I want to integrate it once assuming that $f=f(A^1,A^2,...,A^n)=f(A^\rho)$.

I think the solution should be $\frac{\partial f}{\partial A^\mu} = C_1(A^{\rho\neq\mu}) + (1-\delta_{\mu\nu})\,C_2(A^{\rho\neq\nu})$
where $C_1$ and $C_2$ are functions that have to be determined by initial/boundary conditions, the first term came from the integral wrt $A^\nu$ when $\mu=\nu$ and the second one when $\mu\neq\nu$.

The big confusion arises when I think...
1. Does this really make any sense? - It just looks weird.
2. If $\frac{\partial f}{\partial A^\mu}$ were a covariant tensor, it would make a lot more sense if I had written $h(A_\mu)$ instead of $h(A^\mu)$, wouldn't it?.

 

[haushofer]

Compare to e.g. functions of two variables f(x,y). If one has

$\frac{\partial f}{ \partial x} = 0$

then the solution is given by

$f(x,y) = f(y)$. I.e. the usual integration constant is promoted to an "integration function of y". So I'd say that if in your case

$\frac{\partial g}{ \partial A^{\nu}} = 0$

we get that the solution is a vector field with lower index which can be any function of all the other fields you have; different fields are indepent per definition.