Can Complex Derivatives Clarify Div and Curl Properties?

[Mappe]

In trying to get an intuition for curl and divergence, I've understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z_)), where f(z,z_) is just f(x,y) expressed in z and z conjugate (z_). Is there any way of proving the fundamental properties of div and curl and/or understanding them better by looking at d/dz f(z,z_)?

 

[jambaugh]

I think your complex example will be too limited for a full understanding and proper intuition. Both div and curl have higher dimensional generalizations. To get the best intuitive understanding I would suggest you a.) consider physical applications such as fluid flow and electromagnetic fields, and b.) look at the corresponding integral theorems.