In the introductory page on Green's theorem, we wrote Green's theorem as \begin{align*} \dlint = \iint_\dlr (\curl \dlvf) \cdot \vc{k} \, dA \end{align*} or \begin{align*} \dlint = \iint_\dlr \left( \pdiff{\dlvfc_2}{x} - \pdiff{\dlvfc_1}{y}\right)dA, \end{align*} where $\dlvf(x,y) = (\dlvfc_1(x,y),\dlvfc_2(x,y))$ is a two-dimensional vector field, $\dlr$ is a region in plane, and $\dlc$ is its positively oriented boundary. This notation to ties Green's theorem nicely in with the concept of the curl of a vector field.

We often denote $\dlc$ by $\partial \dlr$ to make it explicit that the curve $\dlc$ is the (positively oriented) boundary of $\dlr$. This notation is also more natural when the region $\dlr$ has more than one boundary component. Then, Green's theorem can look like, for example, \begin{align*} \lint{\partial \dlr}{\dlvf} = \iint_\dlr \left( \pdiff{\dlvfc_2}{x} - \pdiff{\dlvfc_1}{y}\right)dA, \end{align*} Note that the notation $\partial \dlr$ simply means the boundary of $\dlr$. It has nothing to do with a partial derivative.

People frequently write Green's theorem another way. First, they like to change the formula by writing the line integral at the left in terms of components: \begin{align*} \int_{\partial \dlr} F_1\, dx + F_2\, dy = \iint_\dlr \left( \pdiff{\dlvfc_2}{x} - \pdiff{\dlvfc_1}{y}\right)dA. \end{align*} Then, they like to let the vector field be $\vc{F}(x,y) =(P(x,y), Q(x,y))$, so that Green's theorem becomes \begin{align*} \int_{\partial \dlr} P\, dx + Q\, dy= \iint_\dlr \left( \pdiff{Q}{x} - \pdiff{P}{y}\right)dA. \end{align*}

Sometimes, $\vc{F} = (P,Q)$ won't be referred to as a vector field. Instead, one can discuss the above version of Green's theorem as applied to the two scalar valued functions $Q : \dlr \to \R$ and $P : \dlr \to \R$ (confused?).