I was reading Conway's Complex Analysis text and found the proof of the following theorem confusing.
Prove that if $f=u+iv$ is conformal in $G\subseteq\mathbb C$, then the Cauchy-Riemann Equations hold for $u$ and $v$.
So, I worked through the following alternate proof of this theorem and would like to know if my proof is correct.
Let $f=u+iv$ is conformal in $G$. The Jacobian of $f$ is given by $$J_f =\begin{pmatrix} u_x & u_y\\ v_x & v_y \end{pmatrix}.$$ Now, let, $\displaystyle\alpha:=\binom{u_x}{v_x}$ and $\displaystyle\beta:=\binom{u_y}{v_y}$ be the tangent vectors of two smooth curves $\Gamma_1,\Gamma_2$ respectively. By definition, conformal maps preserve angles upon multiplication by $J$. In particular, since $\mathbf{e_1}\perp\mathbf{e_2}$ in $\mathbb R^2$, we have\ $$J(\mathbf{e_1})\perp J(\mathbf{e_2})\implies\alpha\perp\beta.$$ Similarly, $$ J\binom{1}{1}\perp J\binom{1}{-1}\implies (\alpha+\beta)\perp(\alpha-\beta). $$ Using the definition of inner products, we obtain \begin{align*} 0 &= \langle(\alpha+\beta),(\alpha-\beta)\rangle\\ &= \langle\alpha,\alpha\rangle - \langle\beta,\beta\rangle\\ &= ||\alpha||^2 - ||\beta||^2. \end{align*} So, we have $||\alpha||=||\beta||$ and $\alpha\perp\beta$. Since $\alpha,\beta\in\mathbb R^2$, this means $$ \binom{u_x}{v_x}=\pm\binom{v_y}{-u_y}. $$ Now we have $\det(J_f)=\pm\begin{vmatrix} u_x & u_y\\ v_x & v_y \end{vmatrix}=\pm(u_x^2 +u_y^2)$. But since $f$ is conformal, it preserves orientation and $\det(J_f)>0$. So, $\det(J_f)=(u_x^2 +u_y^2)$, and we have $$ \binom{u_x}{v_x}=\binom{v_y}{-u_y}, $$ which are precisely the CR-equations and we are done.
Does everything make sense?
Using the two orthogonality results: $$\alpha \perp \beta\text{ }\text{ }\text{ and }\text{ }\text{ }(\alpha+\beta)\perp(\alpha-\beta)$$ in inner product form gives two equations: $$u_xu_y+v_xv_y =0\text{ }\text{ }\text{ and }\text{ }\text{ }u_x^2 + v_x^2 = u_y^2+v_y^2.$$ Solve for $u_x$ in the first, substitute into the second, solve for $u_y$ (you’ll need quadratic formula for $u_y^2$). The results should give you $C-R$ equations!