Complex form of gauss divergence theorem

Question

Just as complex form of green's theorem $\int {f(z)}dz=i\int\int \frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}dxdy$ where $z=x+iy$ , do we have complex form of gauss divergence theorem ?

Answer 1

In two dimensions Green's theorem and the divergence theorem are basically the same: You get the divergence theorem by applying Green's theorem to a vector field rotated $90^\circ$ at each point. In the same vain you don't get a new theorem when you look at the divergence theorem in a complex disguise.