Which potential-theory tool answers this question?
Green's identities, Green's functions, and the Helmholtz decomposition all sound alike but do different jobs.
GREEN'S IDENTITIES are an integration-by-parts tool, used to prove uniqueness theorems -- reach for them when you need to compare two candidate solutions. A GREEN'S FUNCTION is the fundamental solution for one point source, used to build up a solution by superposition. The HELMHOLTZ DECOMPOSITION is a structural fact about vector fields in general -- reach for it when the question is "split this field into its curl-free and divergence-free parts," not when solving a specific boundary value problem.
Proving uniqueness
For scalar functions u,v with continuous second partials on a solid V with boundary surface S (outward normal \hat{\mathbf n}): \text{(First)} \quad \iiint_V \big(u\,\nabla^2v + \nabla u\cdot\nabla v\big)\,dV = \oiint_S u\,\frac{\partial v} {\partial n}\,dS, \text{(Second)} \quad \iiint_V \big(u\,\nabla^2v - v\,\nabla^2u\big)\, dV = \oiint_S \left(u\,\frac{\partial v}{\partial n} - v\,\frac{ \partial u}{\partial n}\right) dS.
Show proof
First identity: apply the Divergence Theorem (chapter 00) to \mathbf F = u\nabla v. Its divergence is \nabla\cdot(u\nabla v) = \nabla u\cdot\nabla v + u\,\nabla^2v (the ordinary product rule for divergence, chapter 05's index-notation form), and its flux through S is \oiint_S u\nabla v\cdot\hat{\mathbf n}\,dS = \oiint_S u\,\partial v/\partial n\,dS. Setting the Divergence Theorem's two sides equal gives exactly the first identity.
Second identity: apply the first identity twice — once as written, once with u and v swapped — and subtract. The \nabla u\cdot \nabla v terms are identical in both (dot products commute) and cancel, leaving exactly the stated antisymmetric combination.
Building a solution from point sources
Splitting a field into its two structural parts
Example