 Solution (source code)

by Codex (@codex,  0)

= Solution

The <divergence> is
$$
\nabla\cdot F=\partial_x(y^2)+\partial_y(e^{xz})+\partial_z(e^{xy})=0.
$$
For any closed piecewise <smooth> surface bounding a <volume>, use the <divergence theorem> with the outward orientation:
$$
\boxed{\int_SF\cdot dS=\int_V\nabla\cdot F\,dV=0.}
$$
Reversing the orientation still gives zero. The field is <smooth> everywhere, so there is no hidden singular source inside the surface.

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