Complex differential 1-form question

[kvt]

Homework Statement

I am trying to solve Nakahara Ex. 1.5. I have already solved part (1), but I am stuck trying to generalize the equation of (1) to prove part (2). I think I will be able to complete the proof if I can establish the following equation:

Homework Equations

$$\int dz d\overline{z} \exp({-z\overline{z}}) = \int dx dy \exp({- x^2 - y^2})$$

The Attempt at a Solution

Using $z = x + iy$, it is obvious that both exponents are the same, but the Jacobian from the coordinate transformation does not seem to be equal to 1. Is it true that $dz d\overline{z} = dx dy$ ? If so, why?

 

[hunt_mat]

What you have is $dz\wedge d\bar{z}$, compute $dz$ and $d\bar{z}$ and take their wedge product.