📝 题目
例 1 设 $S$ 是球面
$$ {x}^{2} + {y}^{2} + {z}^{2} = {a}^{2}, $$
试计算 $S$ 的面积 $\sigma \left( S\right)$ .
💡 答案与解析
解 我们引入球面的参数方程
$$ \mathbf{r} = \mathbf{r}\left( {\theta ,\varphi }\right) ,\;\left( {\theta ,\varphi }\right) \in D, $$
这里
$$ \mathbf{r}\left( {\theta ,\varphi }\right) = \left( {a\cos \theta \cos \varphi ,a\sin \theta \cos \varphi ,a\sin \varphi }\right) , $$
$$ D = \left\{ {\left( {\theta ,\varphi }\right) \mid 0 \leq \theta \leq {2\pi }, - \frac{\pi }{2} \leq \varphi \leq \frac{\pi }{2}}\right\} . $$
计算得
$$ {\mathbf{r}}_{\theta } = \left( {-a\sin \theta \cos \varphi ,a\cos \theta \cos \varphi ,0}\right) , $$
$$ {\mathbf{r}}_{\varphi } = \left( {-a\cos \theta \sin \varphi , - a\sin \theta \sin \varphi ,a\cos \varphi }\right) , $$
$$ E = {\begin{Vmatrix}{\mathbf{r}}_{\theta }\end{Vmatrix}}^{2} = {a}^{2}{\cos }^{2}\varphi , $$
$$ F = \left( {{\mathbf{r}}_{\theta },{\mathbf{r}}_{\varphi }}\right) = 0, $$
$$ G = {\begin{Vmatrix}{\mathbf{r}}_{\varphi }\end{Vmatrix}}^{2} = {a}^{2}, $$
$$ W = \sqrt{{EG} - {F}^{2}} = {a}^{2}\cos \varphi . $$
所求的面积为
$$ \sigma \left( S\right) = {\iint }_{D}\sqrt{{EG} - {F}^{2}}\mathrm{\;d}\theta \mathrm{d}\varphi $$
$$ = {a}^{2}{\int }_{0}^{2\pi }\mathrm{d}\theta {\int }_{-\pi /2}^{\pi /2}\cos \varphi \mathrm{d}\varphi = {4\pi }{a}^{2}. $$