Complex form of Green's theorem is $\int _{\partial S}{f(z)\,dz}=i\int \int_S{\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\,dx\,dy}$. The following is just my calculation to show both sides equal.
$LHS=\int _{\partial S}{f(z)\,dz}=\int_{\partial S}{(u+iv)(dx+i\,dy)}=\int_{\partial S}{(u\,dx-v\,dy)+i(u\,dy+v\,dx)}$ $RHS=i\int \int_S{\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\,dx\,dy}=i\int \int_S{\frac{\partial f}{\partial x}\,dx\,dy+\int \int_Si\frac{\partial f}{\partial y}\,dx\,dy}=i\int_S{f\,dy-\int \int_Si\frac{\partial f}{\partial y}\,dy\,dx}=i\int_S{f\,dy-\int_Sif\,dx}=i\int_S{(u+iv)\,dy-\int_Si(u+iv)\,dx}=\int_S{i(u+iv)\,dy+\int_S(u+iv)\,dx}=LHS$ Actually what I want to ask here is do we need to add negative sign when change the order of $dx\,dy$ in complex integral?