📝 题目
6. 6.10 设 ${\mathbf{R}}^{3}$ 空间有一变换 $T : {x}_{i} = {x}_{i}\left( {{p}_{1},{p}_{2},{p}_{3}}\right) \left( {i = 1,2,3}\right)$ ,或记作 $\mathbf{x} = T\left( \mathbf{p}\right)$ . 又设向量 $\frac{\partial \mathbf{x}}{\partial {p}_{1}},\frac{\partial \mathbf{x}}{\partial {p}_{2}},\frac{\partial \mathbf{x}}{\partial {p}_{3}}$ 两两互相正交,记 ${H}_{i} = \left| \frac{\partial \mathbf{x}}{\partial {p}_{i}}\right| \left( {i = 1,2,3}\right)$ ,单位向量 ${\mathbf{e}}_{i} = \frac{1}{{H}_{i}}\frac{\partial \mathbf{x}}{\partial {p}_{i}}\left( {i = 1,2,3}\right)$ . 又 $\mathbf{F} = {F}_{1}{\mathbf{e}}_{1} + {F}_{2}{\mathbf{e}}_{2} + {F}_{3}{\mathbf{e}}_{3}$ . 则有
$$ \operatorname{div}\mathbf{F} = \frac{1}{{H}_{1}{H}_{2}{H}_{3}}\mathop{\sum }\limits_{{i = 1}}^{3}\frac{\partial }{\partial {p}_{i}}\left( {{F}_{i}\frac{{H}_{1}{H}_{2}{H}_{3}}{{H}_{i}}}\right) . $$
试利用此公式求下列各式:
(1)在空间柱坐标系,求 $\operatorname{div}\operatorname{grad}u\left( {r,\theta ,z}\right)$ ;
(2)在平面极坐标系,求 $\operatorname{div}\operatorname{grad}u\left( {r,\theta }\right)$ ；
(3)在空间球坐标系,求 $\operatorname{div}\operatorname{grad}u\left( {r,\varphi ,\theta }\right)$ .
💡 答案与解析
例 4 $$ \frac{\partial u}{\partial {\mathbf{e}}_{r}} = \frac{\partial u}{\partial r},\;\frac{\partial u}{\partial {\mathbf{e}}_{\theta }} = \frac{1}{r}\frac{\partial u}{\partial \theta },\;\frac{\partial u}{\partial {\mathbf{e}}_{z}} = \frac{\partial u}{\partial z}. $$
6.6.6 证方向导数 $\frac{\partial u}{\partial {e}_{r}} = \frac{\partial u}{\partial r},\frac{\partial u}{\partial {e}_{\varphi }} = \frac{1}{r}\frac{\partial u}{\partial \varphi },\frac{\partial u}{\partial {e}_{\theta }} = \frac{1}{r\sin \varphi }\frac{\partial u}{\partial \theta }$ .
6.6.7 (1) $v = {\omega l} \times r$ ; (2) rot $v = {2\omega l}$ .
6.6.8 ${xyz}\left( {x + y + z}\right) + c$ .
6.6.10 (1) ${H}_{1} = 1,{e}_{1} = {e}_{r},{H}_{2} = r,{e}_{2} = {e}_{\theta },{H}_{3} = 1,{e}_{3} = {e}_{z}$ ,
$$ \frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial u}{\partial r}}\right) + \frac{1}{{r}^{2}}\frac{{\partial }^{2}u}{\partial {\theta }^{2}} + \frac{{\partial }^{2}u}{\partial {z}^{2}}; $$
(2) $\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial u}{\partial r}}\right) + \frac{1}{{r}^{2}}\frac{{\partial }^{2}u}{\partial {\theta }^{2}}$ ；
(3) ${H}_{1} = 1,{e}_{1} = {e}_{r},{H}_{2} = r,{e}_{2} = {e}_{\varphi },{H}_{3} = r\sin \varphi ,{e}_{3} = {e}_{\theta }$ ,
$$ \frac{1}{{r}^{2}}\frac{\partial }{\partial r}\left( {{r}^{2}\frac{\partial u}{\partial r}}\right) + \frac{1}{{r}^{2}\sin \varphi }\frac{\partial }{\partial \varphi }\left( {\sin \varphi \frac{\partial u}{\partial \varphi }}\right) + \frac{1}{{r}^{2}{\sin }^{2}\varphi }\frac{{\partial }^{2}u}{\partial {\theta }^{2}}. $$