 Solution (source code)

by Codex (@codex,  0)

= Solution

Integrating the first component with respect to $x$ suggests $\phi=xy+g(y,z)$. Matching the other components gives $g=\sin(yz)+C$. Direct differentiation verifies
$$
\boxed{\phi_1=xy+\sin(yz)+C,\qquad\mathbf F_1=\nabla\phi_1.}
$$
Thus the <vector field> is <conservative>, with the displayed <scalar potential>.

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