📝 题目
6.6.6 设 $u = u\left( {x,y,z}\right)$ ,作球坐标变换: $x = r\cos \theta \sin \varphi ,y = r\sin \theta \sin \varphi ,z =$ $r\cos \varphi$ . 令 ${\mathbf{e}}_{r},{\mathbf{e}}_{\varphi },{\mathbf{e}}_{\theta }$ 为两两正交的单位向量. 证明
$$ \operatorname{grad}u = \frac{\partial u}{\partial r}{\mathbf{e}}_{r} + \frac{1}{r}\frac{\partial u}{\partial \varphi }{\mathbf{e}}_{\varphi } + \frac{1}{r\sin \varphi }\frac{\partial u}{\partial \theta }{\mathbf{e}}_{\theta }. $$
💡 答案与解析
6.6.6 证方向导数 $\frac{\partial u}{\partial {e}_{r}} = \frac{\partial u}{\partial r},\frac{\partial u}{\partial {e}_{\varphi }} = \frac{1}{r}\frac{\partial u}{\partial \varphi },\frac{\partial u}{\partial {e}_{\theta }} = \frac{1}{r\sin \varphi }\frac{\partial u}{\partial \theta }$ .