I am looking at the following exercise:
Let $\Phi : U \rightarrow V$ be a diffeomorphism between open subsets of $\mathbb{R}^2$.
Write $$\Phi (u, v)=(f(u, v), g(u, v))$$ where $f$ and $g$ are smooth functions on the $uv$-plane.
Show that $\Phi$ is conformal if and only if
$$\text{ either } (f_u = g_v \text{ and } f_v = −g_u) \text{ or } (f_u = −g_v \text{ and } f_v = g_u)$$
Show that, if $J(\Phi )$ is the Jacobian matrix of $\Phi$, then $\det (J(\Phi)) > 0$ in the first case and $\det (J(\Phi )) < 0$ in the second case.
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I have done the following:
$\Phi$ is conformal if and only if the first funamental form of $\Phi$ is $\lambda (du^2+dv^2$ for some smooth function $\lambda (u,v)$.
We have $$\Phi_u=\Phi_ff_u+\Phi_gg_u \ \text{ and } \ \Phi_v=\Phi_ff_v+\Phi_gg_v$$
$$E=\|\Phi_u\|^2=\|\Phi_ff_u+\Phi_gg_u \|^2=f_u^2\|\Phi_f\|^2+2f_ug_u\Phi_f\cdot \Phi_g+g_u^2\|\Phi_g\|^2\\ F=\Phi_u \cdot \Phi_v=f_uf_v\|\Phi_f\|^2+f_ug_v\Phi_f\cdot \Phi_g+f_vg_u\Phi_f\cdot \Phi_g+g_ug_v\|\Phi_g\|^2\\ G=\|\Phi_v\|^2=\|\Phi_ff_v+\Phi_gg_v \|^2=f_v^2\|\Phi_f\|^2+2f_vg_v\Phi_f\cdot \Phi_g+g_v^2\|\Phi_g\|^2$$
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$\Phi$ is coformal if and only if $E=G$ and $F=0$, so
$$f_u^2\|\Phi_f\|^2+2f_ug_u\Phi_f\cdot \Phi_g+g_u^2\|\Phi_g\|^2=f_v^2\|\Phi_f\|^2+2f_vg_v\Phi_f\cdot \Phi_g+g_v^2\|\Phi_g\|^2 \tag 1$$ $$\text{ and } $$ $$\Phi_u \cdot \Phi_v=f_uf_v\|\Phi_f\|^2+f_ug_v\Phi_f\cdot \Phi_g+f_vg_u\Phi_f\cdot \Phi_g+g_ug_v\|\Phi_g\|^2=0 \tag 2$$
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$$(1) \Rightarrow (f_u^2-f_v^2)\|\Phi_f\|^2+2(f_ug_u-f_vg_v)\Phi_f\cdot \Phi_g+(g_u^2-g_v^2)\|\Phi_g\|^2=0 \tag 3$$
Adding the relations $(2)$ and $(3)$ we get $$(f_u^2+f_uf_v-f_v^2)\|\Phi_f\|^2+(2f_ug_u-2f_vg_v+f_ug_v+f_vg_u)\Phi_f\cdot \Phi_g+(g_u^2+g_ug_v-g_v^2)\|\Phi_g\|^2=0$$
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Is everything correct so far?
How could we continue?
$\Phi$ is conformal is equivalent to saying that for every $x$ the Jacobian matrix is a similitude which is equivalent to saying that:
$Jac(\Phi_x)=\pmatrix{a_x &b_x\cr -b_x & a_x}$ or $Jac(\Phi_x)=\pmatrix{a_x &b_x\cr b_x & -a_x}$ since
$Jac(\Phi_x)=\pmatrix{\partial f_x &\partial f_y\cr \partial g_x & \partial g_y}$ you have the result.