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# 问题

Questions in category: 曲线曲面论 (Curve and surface theory).

## 曲面的Gauss曲率的公式

Posted by haifeng on 2014-08-03 09:38:23 last update 2014-08-03 10:36:33 | Answers (1) | 收藏

$W^4K= \begin{vmatrix} (-\frac{1}{2}G_{uu}+F_{uv}-\frac{1}{2}E_{vv}) & \frac{1}{2}E_u & (F_u-\frac{1}{2}E_v)\\ (F_v-\frac{1}{2}G_u) & E & F\\ \frac{1}{2}G_v & F & G\\ \end{vmatrix}- \begin{vmatrix} 0 & \frac{1}{2}E_v & \frac{1}{2}G_u\\ \frac{1}{2}E_v & E & F\\ \frac{1}{2}G_u & F & G\\ \end{vmatrix}$

$K=\frac{1}{2\lambda^3}(\lambda_u^2+\lambda_v^2-\lambda\Delta\lambda),$

$\Delta=\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2}.$

$\Delta\log\lambda=\frac{\lambda\Delta\lambda-(\lambda_u^2+\lambda_v^2)}{\lambda^2},$

$K=-\frac{1}{2\lambda}\Delta\log\lambda.$

References:

E. F. Beckenbach and T. Rado, Subharmonic functions and surfaces of negative curvature. Trans. Amer. Math. Soc. 35 (1933), 662–674.