Conservative vector field

Recall a vector field is a function $f: {\bf R}^2 \rightarrow {\bf R}^2$. It is called conservative if it is a gradient of a scalar function: $F = {\bf \hspace{3pt} grad \hspace{3pt}} f$.

Theorem. Suppose

$F = (P,Q)$ and $P_y = Q_x$ on region $D$.

Then $F$ is conservative provided $D$ is simply connected.

It is also path-independent.