📝 题目
6. 6.1 设 $u\left( {x,y,z}\right) \in {C}^{2},f\left( t\right) \in {C}^{2}$ . 求
(1) grad $f\left( u\right)$ ; (2) div grad $f\left( u\right)$ .
${6.6.2c}$ 为常向量, $r = \sqrt{{x}^{2} + {y}^{2} + {z}^{2}},f\left( r\right)$ 可微. 求
(1) $\operatorname{div}\left\lbrack {f\left( r\right) c}\right\rbrack$ ; (2) $\operatorname{rot}\left\lbrack {f\left( r\right) c}\right\rbrack$ .
${6.6.3c}$ 为常向量, $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k},r = \left| \mathbf{r}\right|$ . 求
(1) $\operatorname{div}\left\lbrack {\mathbf{c} \times f\left( \mathbf{r}\right) \mathbf{r}}\right\rbrack$ ; (2) rot $\left\lbrack {\mathbf{c} \times f\left( \mathbf{r}\right) \mathbf{r}}\right\rbrack$ .
💡 答案与解析
6. 6.1 (1) ${f}^{\prime }\left( u\right) \nabla u$ ; (2) ${f}^{\prime \prime }\left( u\right) \nabla u \cdot \nabla u + {f}^{\prime }\left( u\right) {\Delta u}$ .