Divergence theorem in complex coordinates This question is related to Stokes' theorem in complex coordinates (CFT)
but, I still don't understand :(
Namely how to prove the divergence theorem in complex coordinate in Eq (2.1.9) in Polchinski's string theory 
$$\int_R d^2 z (\partial_z v^z + \partial_{\bar{z}} v^{\bar{z}})= i \oint_{\partial R} (v^z d \bar{z} - v^{\bar{z}} dz ) (1) $$ 
I may try
$$ \int_R dx dy ( \partial_x F_y - \partial_y F_x)  = \oint_{\partial R} (F_x dx + F_y dy)(2) $$, but what kind of substitution I should use to get Eq. (1)?
 A: Let $\sigma^1$ and $\sigma^2$ be real coordinates on $\mathbb R^2$.  Using the results on page 33, we find that
\begin{align}
    \partial_zv^z
    &= \frac{1}{2}(\partial_1 -i\partial_2)(v^1 + iv^2)
    = \frac{1}{2}(\partial_1v^1 + i\partial_1v^2 - i\partial_2v^1 + \partial_2v^2)  \\
    \partial_{\bar z}v^{\bar z}
    &= \frac{1}{2}(\partial_1 +i\partial_2)(v^1 - iv^2)
    = \frac{1}{2}(\partial_1v^1 - i\partial_1v^2 + i\partial_2v^1 + \partial_2v^2)
\end{align}
and therefore using $d^2z = dz\,d\bar z = 2 d\sigma^1d\sigma^2$
\begin{align}
    \int_R d^2z\,(\partial_zv^z + \partial_{\bar z}v^{\bar z})
    &= 2\int_R d\sigma^1\,d\sigma^2\,(\partial_1v^1 + \partial_2v^2)
\end{align}
similarly, for the right hand side we have
\begin{align}
    v^zd\bar z
    &= (v^1 + iv^2)(d\sigma^1 - id\sigma^2) = v^1d\sigma^1 - iv^1d\sigma^2 + iv^2d\sigma^1 +v^2d\sigma^2 \\
    v^{\bar z}dz
    &= (v^1 - iv^2)(d\sigma^1 + id\sigma^2) = v^1d\sigma^1 + iv^1d\sigma^2 + -iv^2d\sigma^1 +v^2d\sigma^2
\end{align}
so that
\begin{align}
    i\oint_{\partial R} v^z d\bar z - v^{\bar z} d z
    &= i\oint_{\partial R} 2i(v^2d\sigma^1 - v^1 d\sigma^2)
    = 2\oint_{\partial R} (v^1 d\sigma^2 -v^2d\sigma^1)
\end{align}
The identity in Polchinski is obtained by setting the left and right hand sides equal to one another which, in this case, gives
\begin{align}
    \int_R d\sigma^1\,d\sigma^2\,(\partial_1v^1 + \partial_2v^2)
    &=\oint_{\partial R} (v^1 d\sigma^2 -v^2d\sigma^1)
\end{align}
which is precisely Stokes' theorem for a region in $\mathbb R^2$.
