📝 题目
6.6.5 设 $u = u\left( {x,y,z}\right)$ ,作柱坐标变换: $x = r\cos \theta ,y = r\sin \theta ,z = z$ . 令 ${e}_{r}$ , ${\mathbf{e}}_{\theta }$ , ${\mathbf{e}}_{z} = \mathbf{k}$ 为两两正交的单位向量. 证明
$$ \operatorname{grad}u = \frac{\partial u}{\partial r}{\mathbf{e}}_{r} + \frac{1}{r}\frac{\partial u}{\partial \theta }{\mathbf{e}}_{\theta } + \frac{\partial u}{\partial z}{\mathbf{e}}_{z}. $$
💡 答案与解析
6.6.5 根据本节典型例题分析例 4 , 只需证
$$ \frac{\partial u}{\partial {\mathbf{e}}_{r}} = \frac{\partial u}{\partial r},\;\frac{\partial u}{\partial {\mathbf{e}}_{\theta }} = \frac{1}{r}\frac{\partial u}{\partial \theta },\;\frac{\partial u}{\partial {\mathbf{e}}_{z}} = \frac{\partial u}{\partial z}. $$