$\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2=\left|2\frac{\partial u}{\partial z}\right|^2$

Question

I'm reading Complex Analysis by Stein and Shakarchi (page 13). The goal is to prove that $\partial f/\partial z=2\, \partial u/\partial z$ where $f=u+iv$ and $z=x+iy$. The proof in the book states that $$\begin{align}\det J_F (x_0,y_0)&=\frac{\partial u}{\partial x}\frac{\partial v}{\partial y}-\frac{\partial v}{\partial x}\frac{\partial u}{\partial y}\\&=\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2\\&\overset{?}{=}\left|2\frac{\partial u}{\partial z}\right|^2.\end{align}$$ I understand everything except for the last step marked with "?". What I get instead is $$\begin{align}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2&=\frac{\partial u^2}{\partial x^2}+\frac{\partial u^2}{\partial y^2}\\&=\frac{\partial u^2\partial y^2+\partial u^2\partial x^2}{\partial x^2\partial y^2}\\&=\frac{\partial u^2(\partial x^2+\partial y^2)}{\partial x^2\partial y^2}\\&=\frac{\partial u^2\partial z^2}{\partial x^2\partial y^2}\end{align}$$ but I don't understand why should that equal $\left|2\frac{\partial u}{\partial z}\right|^2$.

Answer 1

We define \begin{align} \dfrac{\partial u}{\partial z} &:= \dfrac{1}{2}\left(\dfrac{\partial u}{\partial x} - i \dfrac{\partial u}{\partial y}\right) \end{align} So, by definition of the modulus of a complex number (and the fact that $u$ is real valued), it immediately follows that \begin{align} \left(\dfrac{\partial u}{\partial x}\right)^2 + \left(\dfrac{\partial u}{\partial y}\right)^2 &= \left|2\dfrac{\partial u}{\partial z}\right|^2. \end{align}

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As a side note, we also define \begin{align} \dfrac{\partial u}{\partial \overline{z}} &:= \dfrac{1}{2}\left(\dfrac{\partial u}{\partial x} + i \dfrac{\partial u}{\partial y}\right) \end{align} Hence, it also follows that \begin{align} \left(\dfrac{\partial u}{\partial x}\right)^2 + \left(\dfrac{\partial u}{\partial y}\right)^2 &= \left|2\dfrac{\partial u}{\partial z}\right|^2 = \left|2\dfrac{\partial u}{\partial \overline{z}}\right|^2 . \end{align}