\(\mathrm {Exercise \ \oplus \ Problem } \ 18 \ \)
\(\mathrm {Exercise \ \oplus \ Problem } \ 18 \ \) |
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\( \qquad \)你好，这里是我的个人网站数学分析的每周一题栏目(数学分析每周一题，其中数学分析指的是数学中的分析学, 主要包括微积分，实分析，复分析) \(\qquad \ \)——————Alina Lagrange
求曲面 \[ (x^2+y^2 )^2+z^4=y \] 围成的(立体的)体积.
\(\mathcal{P}roof. \)
考虑坐标变换 $$ \left\{\begin{array}{l} x=\sqrt{r \cos \varphi} \cos \theta \\ y=\sqrt{r \cos \varphi} \sin \theta \\ z=\sqrt{r \sin \varphi} \end{array}\right. $$ 注意到曲面表面上有 $$ r^2=\sqrt{r \cos \varphi} \sin \theta $$ Jacobi行列式 $$ \begin{aligned} \frac{\partial(x, y, z)}{\partial(r, \varphi, \theta)}=& \left|\begin{array}{ccc} \frac{\cos \varphi}{2 \sqrt{r} \cos \varphi} \cos \theta & \frac{\cos \varphi}{2 \sqrt{r} \cos \varphi} \sin \theta & \frac{\sin \varphi}{2 \sqrt{r \sin \varphi}} \\ -\frac{r \sin \varphi}{2 \sqrt{r \cos \varphi}} \cos \theta & -\frac{r \sin \varphi}{2 \sqrt{r \cos \varphi}} \sin \theta & \frac{r \cos \varphi}{2 \sqrt{r \sin \varphi}} \\ -\sqrt{r \cos \varphi} \sin \theta & \sqrt{r \cos \varphi} \cos \theta & 0 \end{array}\right| \\ =&-\frac{r \cos ^2 \varphi}{4 \sqrt{r \sin \varphi}} \sin ^2 \theta-\frac{r \sin ^2 \varphi}{4 \sqrt{r \sin \varphi}} \cos ^2 \theta-\frac{r \sin ^2 \varphi}{4 \sqrt{r \sin \varphi}} \sin ^2 \theta\\& \ \ \ \ -\frac{r \cos ^2 \varphi}{4 \sqrt{r \sin \varphi}} \cos ^2 \theta \\ =& -\frac{1}{4} \sqrt{\frac{r}{\sin \varphi}} \end{aligned} $$ 结合对称性, 注意到 \(y \geq 0\). 记 \(\Omega\) 为其位于第一卦限内的部分 $$ \begin{aligned} V & =4 \iiint_{\Omega} \mathrm{d} x \mathrm{~d} y \mathrm{~d} z \\ & =\int_0^{\frac{\pi}{2}} \mathrm{~d} \theta \int_0^{\frac{\pi}{2}} \frac{\mathrm{d} \varphi}{\sqrt{\sin \varphi}} \int_0^{\sqrt[3]{\cos \varphi \sin ^2 \theta}} \sqrt{r} \mathrm{~d} r \\ & =\frac{2}{3} \int_0^{\frac{\pi}{2}} \sqrt{\cot \varphi} \mathrm{d} \varphi \\ & =\frac{1}{3} \mathrm{~B}\left(\frac{1}{4}, \frac{3}{4}\right) \\ & =\frac{1}{3} \frac{\Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)}{\Gamma(1)} \\ & =\frac{\sqrt{2}}{3} \pi \end{aligned} $$\(\qquad \qquad \)