# 平均曲率

## 定义

${\displaystyle p}$  是曲面 ${\displaystyle S}$  上一点，考虑 ${\displaystyle S}$  上过 ${\displaystyle p}$  的所有曲线 ${\displaystyle C_{i}}$ 。每条这样的 ${\displaystyle C_{i}}$ ${\displaystyle p}$  点有一个伴随的曲率 ${\displaystyle K_{i}}$ 。在这些曲率 ${\displaystyle K_{i}}$  中，至少有一个极大值 ${\displaystyle \kappa _{1}}$ 极小值 ${\displaystyle \kappa _{2}}$ ，这两个曲率 ${\displaystyle \kappa _{1},\kappa _{2}}$  称为 ${\displaystyle S}$ 主曲率

${\displaystyle p\in S}$ 平均曲率是两个主曲率的平均值（斯皮瓦克 1999，第3卷，第2章），由欧拉公式其实也是所有曲率的平均值[3]，故有此名。

${\displaystyle H={1 \over 2}(\kappa _{1}+\kappa _{2})\ .}$

${\displaystyle H={\frac {LG-2MF+NE}{2(EG-F^{2})}}\ ,}$

${\displaystyle H={\frac {1}{n}}\sum _{i=1}^{n}\kappa _{i}\ .}$

${\displaystyle H{\vec {n}}=g^{ij}\nabla _{i}\nabla _{j}X\ ,}$

### 3 维空间中曲面

${\displaystyle 2H=\nabla \cdot {\hat {n}}\ ,}$

{\displaystyle {\begin{aligned}2H&=\nabla \cdot \left[{\frac {\nabla (S-z)}{|\nabla (S-z)|}}\right]\\&=\nabla \cdot \left[{\frac {\nabla S}{\sqrt {1+(\nabla S)^{2}}}}\right]\\&={\frac {\left[1+\left({\frac {\partial S}{\partial x}}\right)^{2}\right]{\frac {\partial ^{2}S}{\partial y^{2}}}-2{\frac {\partial S}{\partial x}}{\frac {\partial S}{\partial y}}{\frac {\partial ^{2}S}{\partial x\partial y}}+\left[1+\left({\frac {\partial S}{\partial y}}\right)^{2}\right]{\frac {\partial ^{2}S}{\partial x^{2}}}}{\left[1+\left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}\right]^{\frac {3}{2}}}}.\end{aligned}}}

${\displaystyle 2H={\frac {\frac {\partial ^{2}S}{\partial r^{2}}}{\left[1+\left({\frac {\partial S}{\partial r}}\right)^{2}\right]^{\frac {3}{2}}}}+{\frac {\frac {\partial S}{\partial r}}{r\left[1+\left({\frac {\partial S}{\partial r}}\right)^{2}\right]^{\frac {1}{2}}}}\ }$

## 流体力学

${\displaystyle H_{f}=(\kappa _{1}+\kappa _{2})\ .}$

Costa 极小曲面示意图

## 注释

1. ^ Dubreil-Jacotin on Sophie Germain. [2008-11-16]. （原始内容存档于2008-02-23）.
2. ^
3. ^ 关于角度的平均值。