1. The Bridge Announced — Two Integral Worlds Joined
Let $D$ be a region in $\mathbb{R}^2$ whose boundary $C$ is a positively oriented, piecewise-smooth, simple closed curve. Let $\mathbf{F}(x,y)=\langle P(x,y),\,Q(x,y)\rangle$ be a vector field whose component functions $P$ and $Q$ have continuous first-order partial derivatives on an open region containing $D$. Then
$$ \oint_{C} P\,dx + Q\,dy \;=\; \iint_{D}\!\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA. $$Unpacking the Hypotheses
Positive Orientation
"Positively oriented" means counterclockwise. The rule: as you walk along $C$ in the direction of traversal, the enclosed region $D$ is always on your left. Think of the standard unit circle — angles increase counterclockwise, and the interior is on your left.
The Integrand
The integrand on the right is the cross-derivative difference: $\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$. This is the two-dimensional scalar curl of $\mathbf{F}$. The order matters: $Q_x$ first, minus $P_y$.
Smoothness
The boundary $C$ must be piecewise smooth (finitely many smooth arcs). The partial derivatives $P_y$ and $Q_x$ must be continuous on an open set containing $D$ — no singularities allowed inside the region. This hypothesis has teeth, as we shall see.