📝 题目
例 1 设 $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k},r = \left| \mathbf{r}\right|$ ,求 $\operatorname{grad}f\left( r\right)$ .
💡 答案与解析
解 令 $u = f\left( r\right) ,\frac{\partial u}{\partial x} = {f}^{\prime }\left( r\right) \frac{x}{r},\frac{\partial u}{\partial y} = {f}^{\prime }\left( r\right) \frac{y}{r},\frac{\partial u}{\partial z} = {f}^{\prime }\left( r\right) \frac{z}{r}$ , 所以
$$ \operatorname{grad}f\left( r\right) = {f}^{\prime }\left( r\right) \frac{r}{r}. $$