= Solution
The components of the <curl> of a <gradient> are differences of mixed second derivatives:
$$
\nabla\times\nabla\phi=(\phi_{zy}-\phi_{yz},\ \phi_{xz}-\phi_{zx},\ \phi_{yx}-\phi_{xy}).
$$
If $\phi$ has continuous second <partial derivatives> in a neighborhood, equality of mixed derivatives makes every component vanish. Thus
$$
\boxed{\nabla\times(\nabla\phi)=0.}
$$
The $C^2$ condition is a sufficient regularity hypothesis; no simple-connectivity assumption is needed for this local identity. Topology matters for the converse assertion that every curl-free field has a global potential.
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