Relationship between partial derivatives for two charts on a Riemann surface

Question

Let's say I have a Riemann Surface $X$. I focus on two coordinate charts $(U_1, \phi_1 = x_1 + iy_1)$ and $(U_2, \phi_2 = x_2 + iy_2)$ on it. On the overlap $U_1 \cap U_2$, do I necessarily have

$\dfrac{\partial (x_1 \circ \phi_2^{-1})} {\partial y_2} = 0$?

This is difficult for me to get my head around.

Answer 1

No, different charts don't have to agree about which direction is "real" and which direction is "imaginary". For instance, taking $X=\mathbb{C}$, you could have $\phi_1(z)=z$ and $\phi_2(z)=iz$ and then $x_1\circ\phi_2^{-1}(x+iy)=y$.