Basic question about index notation

Question

I was trying to make a basic calculation. Suppose that $\partial_{\nu}A^{\nu}=f$ where $f$ is a scalar field. How can i find the expression for $A^{\nu}$. I mean, there is a way to express $A^{\nu}$ as the integral of $f$?

Answer 1

No, as soon as you go to more than $1$ dimension there is not enough information to invert a contraction like this. Indeed, if all you know is that $$ \sum_{i=1}^n \frac{\partial A}{\partial x_i}=f(x_1,\ldots,x_n), $$ then if you find one $A$ satisfying this equation then you can add any $B$ satisfying $$ \sum_{i=1}^n \frac{\partial B}{\partial x_i}=0\qquad(\star) $$ and thereby obtain another $A'=A+B$ satisfying the same equation.

For example, any function of the form $$ a_1x_1+a_2x_2+\cdots+a_{n}x_n $$ where $a_1+a_2+\cdots+a_n=0$ will satisfy $(\star)$. (And many, many, others...)