I am stuck with a math problem that I thought should be straightforward. Maybe I'm missing something here and you can help me.
The key idea is that I have to integrate this function over a volume:
$ I = \int_V \frac{\partial u}{\partial x} dV $
where the volume V is in the x-y plane. I thought of using complex analysis since I do not have a closed form for u, but instead I have an expression for F(z) with
$ F(z) = u + iv \quad \text{with} \quad z=x+iy$
I therefore thought of using complex analysis.
I'll explain my first approach, which I believe was incorrect: First, I thought of the integral "I" as the real part of the whole thing:
$ I = Re\left(\int_V \frac{d F}{d z} dV \right)$
and then use the divergence theorem to cast this not as a volume integral, but as a contour integral.
As you can see already, the approach is wrong in many levels: mostly because dF/dz does not correspond to the divergence (does it?).
I am sure there should be an easy way of converting this integral into a contour integral, but I have been unsuccessful finding it.
Anyone has dealt with this type of problem before?
Many thanks in advance!
Ignacio
Edit:
Ok, after some research I found out that, by defining J as
$J = \displaystyle \frac{i}{2} \int_V \frac{dF}{dz} dz \wedge dz^* $
it is shown that I = Re(J), since $dz\wedge dz^* = -2idxdy$
I believe that J can be calculated using Stokes theorem... work in progress...