So by messing around with some stuff in my own research I came across this problem and I have no idea how to proceed. I suspect it may have something to do with solving systems of PDE's but I could be wrong. For all I know, there may be no analytic way to solve it.
Given a set of functions $ \left\{ f_i(x_1,...,x_n) \right\} $ $i=1,...,n$
Find a set of g's $ \left\{ g_j(x_1,...,x_n) \right\} $ $j=1,...,m$ that satisfy:
$ \begin{pmatrix} \frac{\partial_{g_1}}{\partial_{x_1}}f_1 + & \cdots & + \frac{\partial_{g_1}}{\partial_{x_n}}f_n\\ \frac{\partial_{g_2}}{\partial_{x_1}}f_1 + & \cdots & + \frac{\partial_{g_2}}{\partial_{x_n}}f_n\\ \vdots & & \vdots \\ \frac{\partial_{g_m}}{\partial_{x_1}}f_1 + & \cdots & + \frac{\partial_{g_m}}{\partial_{x_n}}f_n\\ \end{pmatrix} = \begin{pmatrix} G_1(g_1,...,g_m) \\ G_2(g_1,...,g_m) \\ \vdots \\ G_m(g_1,...,g_m) \end{pmatrix} $
Where $ \left\{ G_j \right\} $ are some arbitrary functions expressed in terms of the little g's