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I remember from Folland's PDE book an anecdote about Green convincing himself of the existence of a Green's Function:

Let $\Omega$ be a vacuum and $S$ a perfectly conducting shell grounded to zero potential. Place a negative charge at $x\in \Omega$. This induces a positive charge on the shell $S$. Indeed, the Green's Function $G(x,y)$ is the induced charge at a point $y$.

I would also add that the physical interpretations of gradients, divergence and curl are indespensible for REMEMBERING the various theorems of Gauss, Stokes and Green. For example, the curl of a velocity field is twice the angular acceleration, a fact that facilitates the order of differences of partials in the curl operator.

Let $\Omega$ be a vacuum and $S$ a perfectly conducting shell grounded to zero potential. Place a negative charge at $x\in \Omega$. This induces a positive charge on the shell $S$. Indeed, the Green's Function $G(x,y)$ is the induced charge at a point $y$.