This relates the line integral over a boundary of a region with the double integral over the interior of the region.
A closed curve is positively oriented if it is being traversed counter-clockwise.
For a positively oriented, piecewise-smooth, simple closed planar curve \(C\) with interior region \(R\) if \(P\) and \(Q\) have continuous partial derivatives on some open neighborhood containing \(R,\) then \[ \int\limits_C P\,\mathrm{d}x + Q\,\mathrm{d}y = \iint\limits_R \bigg(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\bigg)\,\mathrm{d}A \]
A short-hand notation for the boundary of a region \(R\) is \(\partial R\) which is a brilliant overload of the partial-differential operator in light of Stokes’ Theorem.
When the orientation of the curve \(C\) needs to be acknowledged we use the notation \[\oint\limits_C P\,\mathrm{d}x + Q\,\mathrm{d}y\] and sometimes even put a little arrow on that circle in \(\oint\) to specify the orientation.
We can extend Green’s theorem to non-simple closed regions.