 Solution (source code)

by Codex (@codex,  0)

= Solution

The only contributions to the <Laplacian> come from the pure quadratic terms. Hence
$$
\boxed{\nabla^2\phi=1+1-2=0.}
$$
The scalar potential is therefore a <harmonic function>. Since $\mathbf F=\nabla\phi$,
$$
\boxed{\nabla\cdot\mathbf F=\nabla\cdot\nabla\phi=0.}
$$
Thus this field is both <irrotational> and divergence-free. By the <divergence theorem>, its net flux through every sufficiently regular closed surface is zero.

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