📝 题目
例 4 设 $u = u\left( {x,y}\right)$ ,令 $x = r\cos \theta ,y = r\sin \theta ,u = u(r\cos \theta$ , $r\sin \theta )$ . 再令 ${\mathbf{e}}_{r},{\mathbf{e}}_{\theta }$ 分别是 $M$ 点的径向与圆周方向的单位向量. 证明: 在 $M$ 点有
$$ \operatorname{grad}u = \frac{\partial u}{\partial r}{e}_{r} + \frac{1}{r}\frac{\partial u}{\partial \theta }{e}_{\theta }. $$
💡 答案与解析
证 由方向导数的计算公式和复合函数求导法则, 得
$$ \frac{\partial u}{\partial {e}_{r}} = \frac{\partial u}{\partial x}\cos \theta + \frac{\partial u}{\partial y}\sin \theta = \frac{\partial u}{\partial r}, $$
$$ \frac{\partial u}{\partial {e}_{\theta }} = \frac{\partial u}{\partial x}\left( {-\sin \theta }\right) + \frac{\partial u}{\partial y}\cos \theta = \frac{1}{r}\frac{\partial u}{\partial \theta }. $$
应用