Conformal map

Definition

Let $U$ be an open subset of the complex plane, identified with $\mathbb{R}^2$, and let $z_0 \in U$. Let $f : U \to \mathbb{C}$ be differentiable at $z_0$ as a map from an open subset of $\mathbb{R}^2$ to $\mathbb{R}^2$, with Jacobian matrix $J$ at $z_0$. The map $f$ is conformal at $z_0$ if $J = \lambda Q$ for a real number $\lambda > 0$ and a rotation matrix $Q$. A rotation matrix is an orthogonal $2 \times 2$ matrix with determinant $1$. The map $f$ is conformal on $U$ if it is conformal at each point of $U$.

Remarks

Equivalently, $f$ is conformal at $z_0$ when its real derivative at $z_0$ is the map $h \mapsto a h$ for a complex number $a \neq 0$. A map that is conformal at $z_0$ preserves the angles between curves through $z_0$, and their orientation. Some authors also allow orientation-reversing maps, such as $z \mapsto \bar{z}$, whose Jacobian is a positive number times an orthogonal matrix. Other authors use "conformal map" for an injective holomorphic map on an open set.

Explanations

L1 human

A conformal map can stretch and turn a small figure, but it does not distort its shape. If two curves cross at an angle of $30$ degrees, then their images cross at the same angle, in the same direction. For example, $z \mapsto 2z$ and $z \mapsto iz$ are conformal at each point.