Conformal Mapping preserves the angles between the smooth curves.

Question

I am interested to proof that conformal mapping preserves the angles between the smooth curves. I will be greatful if anyone can help me in it.

Answer 1

The definition of a conformal map is one that preserves angles.

A conformal map in the complex analytical sense is any holomorphic function on a subset $U \subset \mathbb{C}$ with non-zero derivative on $U$.

Equivalently, a conformal map is a mapping whose Jacobian matrix is a scaled multiple of a rotation matrix.

So, consider some analytic function $f=u+iv$. It's Jacobian is

$$\begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{pmatrix} = \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ -\frac{\partial u}{\partial y} & \frac{\partial u}{\partial x} \end{pmatrix}$$ and the determinant is

$$\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2 = C$$

as long as its derivative is not zero.

It should be clear that this is proportional to

$$\det \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{pmatrix}$$

by a factor of a non-zero scalar $C$. You can also more directly relate $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ to $\cos \theta$, $\sin \theta$ respectively.