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[tex]\int\int_{D}f(x,y)\frac{\partial g}{\partial x,y}dA=\oint_{\Sigma}f(x,y)g(x,y)\mathbf{n}\cdot d\mathbf{s}-\int\int_{D}g(x,y)\frac{\partial f}{\partial x,y}dA[/tex]

For lack of a better notation, I used [tex]\frac{\partial f}{\partial x,y}[/tex] to represent the fact that the derivative could be with respect to either x or y.

Thanks for your help.