We know that given the divergence and curl of a vector field (and appropriate boundary conditions) it is possible to construct a unique vector field in $\mathbb R^3$. The specific problem I am thinking about is related to the PDE $$\operatorname{div} F = g,$$ where $F \colon \mathbb R^n \to \mathbb R^n$ is a vector field and $g \colon \mathbb R^n \to \mathbb R$ is a scalar field, and $\operatorname{div}$ is the $n$-dimensional generalization of the divergence given by $$\operatorname{div} F = \frac{dF_{i}}{dx_{i}}$$ (summation implied). What additional pieces of information are necessary to uniquely specify $F$ given the function $g$ (we know the answer is the curl of $F$ in 3D)?
Higher Dimensional Generalization of Helmholtz Theorem
2
$\begingroup$
pde
vector-analysis