 Solution

For $\mathbf{F}=\nabla \varphi$, the definition of the Laplacian gives $\nabla \cdot \mathbf{F}={\nabla }^2\varphi =f$. Integrating over $V$ and applying the divergence theorem yieldsThis is the flux form of Poisson equation, for an outward-oriented piecewise smooth boundary and the usual smoothness of $\varphi$ on a neighborhood of the volume. In particular, a singularity cannot be ignored when applying this identity.