= Solution
Regard the planar <vector field> as $(A(x,y),B(x,y),0)$, with no dependence on $z$. Its <curl> is
$$
\nabla\times\mathbf V=
\begin{vmatrix}
\mathbf i&\mathbf j&\mathbf k\\
\partial_x&\partial_y&\partial_z\\
A&B&0
\end{vmatrix}
=\left(\frac{\partial B}{\partial x}-\frac{\partial A}{\partial y}\right)\mathbf k.
$$
Thus, for continuously differentiable $A$ and $B$, the necessary and sufficient pointwise condition is
$$
\boxed{\frac{\partial B}{\partial x}=\frac{\partial A}{\partial y}.}
$$
To deduce global <path independence> from this local condition also requires a suitable domain, for example a <simply connected domain>.
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