📂Partial Differential Equations

Exterior Unit Normal Vector

Definition¹

Let $U\subset \mathbb{R}^{n}$ be an open set. Let the boundary of $U$ be $\partial U$, which is a $\partial U \in C^1$. Then, the following outward unit normal vector can be defined:

$$ \boldsymbol{\nu}=(\nu^{1}, \nu^{2}, \dots, \nu^{n}) \quad \text{and} \quad |\boldsymbol{\nu}|=1 $$

$\boldsymbol{\nu}$ is a vector that touches a point on the boundary, has a magnitude of 1, and points outward. Let it be $u \in C^{1}(\bar{U})$. Then, the directional derivative $\dfrac{\partial u}{\partial \nu}$ is defined as follows:

$$ \dfrac{\partial u}{\partial \nu} := \boldsymbol{\nu} \cdot Du=(\nu^1,\cdots,\nu^n)\cdot(u_{x_{1}}, \cdots, u_{x_{n}}) $$

$D=D^{1}$ is the multi-index notation, and $Du$ is the gradient of $u$.