Green's Theorem has the formula $$ \int_C Mdx+Ndy=\int\int_D\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy $$
There is also a well known test for finding an integrating factor for a form $Mdx+Ndy$ depending on whether $\frac{1}{M}\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)$ or $\frac{1}{N}\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)$ depend only on $x$ or $y$.
Is there a deeper relationship between the $\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)$ showing up in both things, or is it coincidence?