Regarding higher-dimensional space. For a Riemannian manifold M of any dimension, and a smooth vector field V, we can pose the problem: find smooth U:M→R that minimizes ∫M||∂U(x)−V(x)||2μ(dx), where μ is the canonical measure on M induced by the metric. If either M is compact or we impose appropriate boundary conditions on U and V, then I’m pretty sure this equivalent to solving the elliptic differential equation ΔU=∂⋅V. Here, the Laplacian and ∂V are defined using the Levi-Civita connection. If M is connected then, under these conditions, the equation has a unique solution up to an additive constant.
Regarding higher-dimensional space. For a Riemannian manifold M of any dimension, and a smooth vector field V, we can pose the problem: find smooth U:M→R that minimizes ∫M||∂U(x)−V(x)||2μ(dx), where μ is the canonical measure on M induced by the metric. If either M is compact or we impose appropriate boundary conditions on U and V, then I’m pretty sure this equivalent to solving the elliptic differential equation ΔU=∂⋅V. Here, the Laplacian and ∂V are defined using the Levi-Civita connection. If M is connected then, under these conditions, the equation has a unique solution up to an additive constant.