Higher Dimensional Generalization of Helmholtz Theorem 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)?
 A: The appropriate generalization you want is to consider the vector field $F$ and the scalar field $g$ as $(n-1)-$ and $n$-forms in the deRham complex.  That is
$$ F \in \Omega^{n-1}(\mathbb{R}^n) = \Omega^{n-1} \otimes C^\infty(\mathbb{R}^n) , $$ $$F = f_1 \ dx_2 \wedge ... \wedge dx_n + f_2 \  dx_1 \wedge dx_3 \wedge ... \wedge dx_n + $$ $$ f_3 \ dx_1 \wedge dx_2 \wedge dx_4 \wedge ... \wedge dx_n + ...+ f_n \ dx_1 \wedge ... \wedge dx_{n-1} \wedge dx_n  $$
and
$$ g \in \Omega^n(\mathbb{R}^n) $$ $$ g = G \ dx_1 \wedge dx_2 \wedge ... \wedge dx_n $$
Framing it in these terms the equation
$$ div( F) = g $$
is $$dF = g $$ where d is the external derivative $$ d : \Omega^j(\mathbb{R}^n) \rightarrow \Omega^{j+1}(\mathbb{R}^n) $$
Now your question becomes: given $ g$, what $F$ satisfy $dF = g$.  The answer is certainly there is no unique function/n-1 form $F$ which satisfies this equation.  Given any such solution $F_0$, then $F = F_0 + dE$ is also a solution where $E$ is a $(n-2)-$form, because $d^2 = 0$.  Further, given any such $g$ there must exist at least one $F$ because $g$ is a closed $n$-form, meaning $dg = 0$ by the Poincare theorem, every closed $n$-form is also exact, meaning there exists such a $(n-1)-$form $F$ such that $dF = g$.  (In the language of deRham cohomology, $H^n_{DR}(\mathbb{R}^n) = 0$.)
The more interesting question is this: what happens if we look now not at the deRham complex on $\mathbb{R}^n$ but on subsets of $\mathbb{R}^n$ that are topologically non-trivial?  Or on $n$-dimensional manifolds? Are there then $n$-forms $g$ for which there are no $F$ such that $dF = g$?
In other words, the existence of closed $n$-forms that are not exact will depend on the topology of the underlying space, and vice versa.  It's a great topic.
A: I read that W. Hauser generalized Helmholtz's theorem to R⁴ by proving it for second-rank tensors [1,2]. but unfortunately I could not found his original papers for download (to see the details).. tray by yourself to found these references, you may be more lucky than me:
[1] W. Hauser, "On the Fundamental Equations of Electromagnetism," Am. J. Physics, vol. 38, no. 1, pp. 80-85, 1970.
[2] W. Hauser, Introduction to the Principles of Electromagnetism. Addison-Wesley Educational Publishers Inc, 1971.
