曲面的面积
曲面的面积元与面积
设曲面 \Sigma 由参数方程 x = x(u, v), y = y(u, v), z = z(u, v) 确定，令 Gauss 系数
\begin{cases}
E = \left( \frac{\partial x}{\partial u} \right)^2 + \left( \frac{\partial y}{\partial u} \right)^2 + \left( \frac{\partial z}{\partial u} \right)^2; \\
F = \frac{\partial x}{\partial u} \frac{\partial x}{\partial v} + \frac{\partial y}{\partial u} \frac{\partial y}{\partial v} + \frac{\partial z}{\partial u} \frac{\partial z}{\partial v};\\
G = \left( \frac{\partial x}{\partial v} \right)^2 + \left( \frac{\partial y}{\partial v} \right)^2 + \left( \frac{\partial z}{\partial v} \right)^2.
\end{cases}
则称曲面的面积元为：
\mathrm{d} S = \sqrt{EG - F^2} \mathrm{d} u \mathrm{d} v.
曲面的面积为
S = \iint_D \sqrt{EG - F^2} \mathrm{d} u \mathrm{d} v.
当曲面由二元函数 z = f(x, y) 给出时，可以把 x, y 本身看作曲面的参数方程，则有
E = 1 + \left( \frac{\partial z}{\partial x} \right)^2, \quad
F = \frac{\partial z}{\partial x} \frac{\partial z}{\partial y}, \quad
G = 1 + \left( \frac{\partial z}{\partial y} \right)^2;
于是有
\mathrm{d} S = \sqrt{EG - F^2} \mathrm{d} x \mathrm{d} y = \sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2} \mathrm{d} x \mathrm{d} y.
这个公式有较为明显的几何意义。
当曲面由隐函数 f(x, y, z) = 0 确定时，计算公式为
\mathrm{d} S = \frac{\sqrt{f_x^2 + f_y^2 + f_z^2}}{|f_z|} \mathrm{d} x \mathrm{d} y = \frac{\| \nabla f \|}{|f_z|} \mathrm{d} x \mathrm{d} y.
BEYOND THE ARTICLE
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