While trying to solve a problem involving vector potentials I ran into this type of function: $$ f(x,y,z) = g(x,y,z) + \int \frac\partial{\partial z} \left( \int \frac\partial{\partial y} \left( \int \frac{\partial f}{\partial x}dz \right)dx \right)dy $$
My primary question is if there's a way to simplify/evaluate the triple integral. Secondly, I was wondering if there's a name or theorem or any better way to describe this type of integral.
Thanks to anyone that's able to help.
Edit: Changed title to something that may be more relevant to the question.
Yes, this can be simplified, by interchanging the order of differentiation and integration. For the innermost integral, $$ \frac{\partial}{\partial y}\int \frac{\partial f}{\partial x}\ dz=\int \frac{\partial^2 f}{\partial y\ \partial x}\ dz, $$ and doing the same once more yields that $$ \int \frac{\partial}{\partial z}\int \frac{\partial}{\partial y}\int \frac{\partial f}{\partial x}\ dz\ dx\ dy=\int\int\int \frac{\partial^3 f}{\partial z\partial y\partial x}\ dz\ dx\ dy. $$ Note also that $x,y,z$ can be reordered in any of the $3!$ desired orderings, in both the derivatives and in the integrals, each independently of the other, without affecting the value of the expression. (All this follows from Leibniz' integral rule and equality of mixed partials, which require some technical analytical assumptions about the function $f(x,y,z)$ that are usually satisfied in practice.)