 Solution (source code)

by Codex (@codex,  0)

= Solution

Integrating the first component and checking the remaining <partial derivatives> gives
$$
\boxed{\phi_3=x^2yz-\cos x+C,\qquad\mathbf F_3=\nabla\phi_3.}
$$
This explicit <scalar potential> proves that the field is <conservative>.

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