📂Vector Analysis

Conformal Mapping

Definition¹

Assuming the mapping $\mathbf{f} : \mathbb{R}^{n} \to \mathbb{R}^{m}$ is given as follows.

$$ \mathbf{f}(\mathbf{x}) = \left( f_{1}(\mathbf{x}), f_{2}(\mathbf{x}), \dots, f_{m}(\mathbf{x}) \right),\quad \mathbf{x}\in \R^{n} $$

The total derivative, or Jacobian matrix of $\mathbf{f}$ is as follows.

$$ \mathbf{f}^{\prime} = J = \begin{bmatrix} \dfrac{\partial f_{1}}{\partial x_{1}} & \cdots & \dfrac{\partial f_{1}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_{m}}{\partial x_{1}} & \cdots & \dfrac{\partial f_{m}}{\partial x_{n}} \end{bmatrix} $$

If at every point $\mathbf{x} \in \R^{n}$, the rank of the Jacobian matrix of $\mathbf{f}$ is $n$, then $\mathbf{f}$ is called a regular mapping.