This is a question in Complex Analysis for Mathematical Science and Engineering by Saff and Snider. It's on pg 62.
Question: The Jacobian of a mapping $u = u(x,y)$ $v = v(x,y)$ from the xy-plane to the uv-plane is defined to be the determinant $$ J(x_0,y_0) := \begin{vmatrix} \frac{\partial{u}}{\partial{x}} && \frac{\partial{u}}{\partial{y}} \\ \frac{\partial{v}}{\partial{x}} && \frac{\partial{v}}{\partial{y}} \end{vmatrix}$$
Where the derivatives are all evaluated at $(x_0, y_0)$. Show that if $f = u + iv$ is analytic at $z_0 = x_0 + iy_0$, then $J(x_0,y_0) = |f'(z_0)|^2$.
My attempt to answer: Given f'(z) is analytic $f'(z) = \frac{\partial{u}}{\partial{x}} + i\frac{\partial{v}}{\partial{x}}$ which leads us to $|f'(z_0)|^2 = \frac{\partial{u}}{\partial{x}}^2 - \frac{\partial{v}}{\partial{x}}^2$. We can now use the Cauchy-Riemann equations as substitutions; $$|f'(z_0)|^2 = \frac{\partial{u}}{\partial{x}}\left(\frac{\partial{y}}{\partial{y}} \right)- \frac{\partial{v}}{\partial{x}}\left(-\frac{\partial{v}}{\partial{x}}\right) $$
But now the minus signs in the second term cancel and it is positive. The second term from the determinant should be negative. What did I do wrong?
