$
\Omega \subset \mathbb{R}^2, f: \Omega \rightarrow \mathbb{R}^2
$

$$
f(x, y) = (u(x, y), v(x, y))
$$

$$
J(x, y) =
\begin{bmatrix}
\frac{\partial u}{\partial x}& \frac{\partial u}{\partial y} \\
\frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}
\end{bmatrix}
$$

$
(x, y) \in \Omega, J(x, y) = s(x, y)R(x, y)
$

$s$ is a non-zero scalar.

$R$ is a $2 \times 2$ rotation matrix.

$\mathcal{prop.}$
  1. $f: \Omega \rightarrow \mathbb{R}^2 \text{ and } g: f(\Omega) \rightarrow \mathbb{R}^2$ are conformal map, then $g \circ f$ is conformal map
  2. $f: \Omega \rightarrow \mathbb{R}^2$ is conformal map, $f^{-1}$ is conformal map