📝 题目
例 2 试写出 $\nabla$ 与 $\Delta$ 的柱坐标表示.
💡 答案与解析
解 我们知道,联系直角坐标(x, y, z)与柱坐标 $\left( {r,\theta ,z}\right)$ 的变换公式是
$$ \left\{ \begin{array}{l} x = r\cos \theta , \\ y = r\sin \theta , \\ z = z. \end{array}\right. $$
计算柱坐标的拉梅系数得:
$$ {h}_{1} = \sqrt{{\left( \frac{\partial x}{\partial r}\right) }^{2} + {\left( \frac{\partial y}{\partial r}\right) }^{2} + {\left( \frac{\partial z}{\partial r}\right) }^{2}} = 1, $$
$$ {h}_{2} = \sqrt{{\left( \frac{\partial x}{\partial \theta }\right) }^{2} + {\left( \frac{\partial y}{\partial \theta }\right) }^{2} + {\left( \frac{\partial z}{\partial \theta }\right) }^{2}} = r, $$
$$ {h}_{3} = 1\text{ . } $$
对于数量值函数 $u = u\left( {r,\theta ,z}\right)$ 与向量值函数
$$ \mathbf{U} = {u}_{1}\left( {r,\theta ,z}\right) {\mathbf{e}}_{r} + {u}_{2}\left( {r,\theta ,z}\right) {\mathbf{e}}_{\theta } + {u}_{3}\left( {r,\theta ,z}\right) {\mathbf{e}}_{z} $$
我们有
$$ \nabla u = \frac{\partial u}{\partial r}{\mathbf{e}}_{r} + \frac{1}{r}\frac{\partial u}{\partial \theta }{\mathbf{e}}_{\theta } + \frac{\partial u}{\partial z}{\mathbf{e}}_{z}, $$
$$ \nabla \cdot \mathbf{U} = \frac{1}{r}\frac{\partial }{\partial r}\left( {r{u}_{1}}\right) + \frac{1}{r}\frac{\partial {u}_{2}}{\partial \theta } + \frac{\partial {u}_{3}}{\partial z}, $$
$$ \nabla \times \mathbf{U} = \frac{1}{r}\left| \begin{matrix} {\mathbf{e}}_{r} & r{\mathbf{e}}_{\theta } & {\mathbf{e}}_{z} \\ \frac{\partial }{\partial r} & \frac{\partial }{\partial \theta } & \frac{\partial }{\partial z} \\ {u}_{1} & r{u}_{2} & {u}_{3} \end{matrix}\right| , $$
$$ {\Delta u} = \frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial u}{\partial r}}\right) + \frac{1}{{r}^{2}}\frac{{\partial }^{2}u}{\partial {\theta }^{2}} + \frac{{\partial }^{2}u}{\partial {z}^{2}}. $$