I am interested in whether each component of a divergenceless vector can itself be written as a divergence. My motivation for this question is the characterization of so-called *trivial conservation laws* in physics. I suspect this representation is general, but I have not made much progress in a proof.


**Overview**


Consider the following differential equation for a vector $M$ depending on a set of independent variables $x=(x\_1,x\_2,...,x\_n)$ and an arbitrary smooth vector field $u=(u^1,u^2,...,u^p)$ and its derivatives $\partial u=(u^1\_1,u^1\_2,...,u^p\_n)$:
\begin{align}\label{1}\tag{1}
\sum\_{i=1}^nD\_iM^i(x,u,\partial u(x),...,\partial^m u(x))=0.
\end{align}
Here, $D\_i=\partial\_i+u^j\_i\partial\_{u^j}+...$ denotes a "total derivative", and $\partial^ku$ denotes the set of $k$-th order derivatives of $u$.


**Conjecture**:


The general solution to \eqref{1} for $M$ is as follows:
\begin{align}\label{2}\tag{2}
M^i=\sum\_{j\neq i}D\_jN^{ij}(x,u,\partial u,...,\partial^q u),\quad 1\le i\le n,
\end{align}
where each $N^{ij}=-N^{ji}$ is some function in an anti-symmetric $n\times n$ matrix, and $q\ge m-1$ is some integer.


By "general solution", I mean that if \eqref{1} has a solution $M$, then there exist functions $N$ such that \eqref{2} holds.


**Progress**


It is clear that \eqref{2} is a *particular solution* to \eqref{1}. That it is the general solution is straightforward to derive when $m=1$. We can do this by expanding \eqref{1} using the chain rule and equating coefficients of higher order derivatives to zero. However, this problem quickly becomes intractable as $m$ is increased. I am wondering if there is a more elegant and general way to approach this problem.



General solution to null-divergence equation
The answer to your question is 'yes', that is the general solution. This is one of the basic results in the theory of the variational bicomplex. It is a statement of the vanishing of a certain cohomology group in the variational bicomplex. A good place to look for the proof would be in say, these [lectures](http://digitalcommons.usu.edu/mathsci_facpub/32/) by Ian Anderson on the variational bicomplex.


