Solution
= Solution
Write $f=u+iv$ and $z=x+iy$. The <Cauchy-Riemann equations> give $u_x=v_y$ and $u_y=-v_x$. Thus the real <Jacobian determinant> is
$$
\boxed{\frac{\partial(u,v)}{\partial(x,y)}=u_xv_y-u_yv_x=u_x^2+v_x^2=|u_x+iv_x|^2=|f'(z)|^2.}
$$
This identity also holds at points where $f'(z)=0$; there the Jacobian is zero and local angle preservation is not guaranteed.