A real function $f(x)$ is defined on $N$-dimensional real space where $N \ge 3$. $f(x)$ is differentiable and its gradient with respect to x is $g(x)$. So $g(x)$ is a vector field.
Assume we do not know the closed form of $f(x)$ or $g(x)$. Instead we only know the value of $f(x)$ (real numbers) on one set of data samples $x_0,x_1,\ldots$, and the value of $g(x)$ (real vectors) on a (possibly separate) set of data samples $x_0',x_1',\ldots$.
Under what condition, is it possible to derive a relationship between the function samples $f(x_0), f(x_1), \ldots$ and the gradient samples $g(x_0'), g(x_1'), \ldots$?
Intuitively, the relationship between a function and its gradient is governed by vector calculus results like Green's identities. But such results are only applicable when $f(x)$ and $g(x)$ are available in closed form. It is unclear how to apply it to the case of sampling.
