I am reading the visual complex analysis book and currently the chapter about an concept called "amplitwist". I read that the complex mapping like $$ z \mapsto z^2 $$ $$ z = re^{i\theta} $$
can be expressed locally as an amplitwist (amplification + rotation) which can also be encoded in Jacobian matrix like
$$\begin{bmatrix} \frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}\end{bmatrix}$$
so I calculate $z^2$ as $$ z^2 = r^2 e^{2i\theta} = r^2(\cos(2\theta) + i\sin(2\theta)) $$ $$ u = r^2 \cos(2\theta) $$ $$ v = r^2 \sin(2\theta) $$ therefore $\partial_x u = 2r \cos(2\theta) + r^2 (\cos(2\theta)\frac{d}{dx}) = 2r \cos(2\theta)$
so it means the first element of jacobian matrix should have an argument of $2\theta$ but in the book it is just $\theta$
What I do wrong so I got this argument doubled?