How do you explain the following equations geometrically?

Question

Let D be the field in C, $f=u+iv \in C^{1}(D)$. Proof: $$\begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\newline \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}=\vert \frac{\partial f}{\partial z} \vert^{2} - \vert \frac{\partial f}{\partial \bar{z}} \vert^{2}.$$ especially, when $f \in H(D)$, have $$\begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\newline \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}=\vert f^{'} \vert^{2}.$$ The solution of the problem is easy, but what is the geometric meaning of these two equations? The right-hand side of the equation is hard for me to handle.