矢量分析中的柱坐标与球坐标变换

XantC

本文总目标：推导柱坐标、球坐标下梯度、散度、旋度的表达式。

利用拉梅系数把正交曲线坐标系统一起来的做法，请看：。本篇的解法是最直接的基矢变换。

基变换和坐标变换

在笛卡尔坐标下，微分算子的定义是\(\nabla=\boldsymbol{e_x}\frac{\partial}{\partial x}+\boldsymbol{e_y}\frac{\partial}{\partial y}+\boldsymbol{e_z}\frac{\partial}{\partial z}\)。这个算子依赖于笛卡尔坐标，即一个固定的正交单位矩阵，所以想表示其他坐标（基，或矩阵）下的算子，需要先把新的基转为\(\boldsymbol{e_x},\boldsymbol{e_y},\boldsymbol{e_z}\)的固定基，才能得到正确的结果。

\[ 柱坐标： \begin{bmatrix} \boldsymbol{e_r}\\ \boldsymbol{e_\theta}\\ \boldsymbol{e_z} \end{bmatrix}=\begin{bmatrix} \cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1 \end{bmatrix} \begin{bmatrix} \boldsymbol{e_x}\\ \boldsymbol{e_y}\\ \boldsymbol{e_z} \end{bmatrix} \]

\[ 球坐标： \begin{bmatrix} \boldsymbol{e_r}\\ \boldsymbol{e_\theta}\\ \boldsymbol{e_\phi} \end{bmatrix}=\begin{bmatrix} \sin\theta\cos\phi&\sin\theta\sin\phi&\cos\theta\\ \cos\theta\cos\phi&\cos\theta\sin\phi&-\sin\theta\\ -\sin\phi&\cos\phi&0\\ \end{bmatrix} \begin{bmatrix} \boldsymbol{e_x}\\ \boldsymbol{e_y}\\ \boldsymbol{e_z} \end{bmatrix} \]

令新基为\(\boldsymbol{e}\)，笛卡尔基为\(\boldsymbol{e_{xyz}}\)，记：

\[ \boldsymbol{e}=J\boldsymbol{e_{xyz}} \]

令新坐标（球坐标为例）、笛卡尔坐标为：

\[\boldsymbol{A}=\begin{bmatrix} A_r\\A_\theta\\A_\phi \end{bmatrix},\;\boldsymbol{A_{xyz}}=\begin{bmatrix} A_x\\A_y\\A_z \end{bmatrix}\]

由坐标变换前后向量不变：

\[ \boldsymbol{v}=\boldsymbol{A}^\mathrm{T}\boldsymbol{e} \]

可推得相应的坐标为：

\[ A=JA_{xyz} \]

\[ A_{xyz}=J^\mathrm{T}A \]

有了这个关系，我们就可以任意地变换坐标了。

梯度

\[ 柱坐标：\nabla f=\boldsymbol{e_r}\frac{\partial f}{\partial r}+\boldsymbol{e_\theta}\frac{1}{r}\frac{\partial f}{\partial \theta}+\boldsymbol{e_z}\frac{\partial f}{\partial z} \]

\[ 球坐标：\nabla f=\boldsymbol{e_r}\frac{\partial f}{\partial r}+\boldsymbol{e_\theta}\frac{1}{r}\frac{\partial f}{\partial \theta}+\boldsymbol{e_\phi}\frac{1}{r \sin\theta}\frac{\partial f}{\partial \phi} \]

以柱坐标为例

\[ \begin{align*} \nabla f&=\begin{bmatrix} \frac{\partial f}{\partial x}&\frac{\partial f}{\partial y}&\frac{\partial f}{\partial z} \end{bmatrix}\begin{bmatrix} \boldsymbol{e_x}\\ \boldsymbol{e_y}\\ \boldsymbol{e_z} \end{bmatrix}\\ &=\begin{bmatrix} \frac{\partial f}{\partial x}&\frac{\partial f}{\partial y}&\frac{\partial f}{\partial z} \end{bmatrix}J^\mathrm{T}\begin{bmatrix} \boldsymbol{e_r}\\\boldsymbol{e_\theta}\\\boldsymbol{e_z} \end{bmatrix}\\ &=\begin{bmatrix} \frac{\partial f}{\partial x}&\frac{\partial f}{\partial y}&\frac{\partial f}{\partial z} \end{bmatrix}\begin{bmatrix} \boldsymbol{e_r}\cos\theta-\boldsymbol{e_\theta}\sin\theta\\ \boldsymbol{e_r}\sin\theta+\boldsymbol{e_\theta}\cos\theta\\\boldsymbol{e_z} \end{bmatrix} \end{align*} \]

计算柱坐标下的\(\frac{\partial f}{\partial x}\)和\(\frac{\partial f}{\partial y}\)：

\[ \begin{align*} \frac{\partial f(r,\theta, z)}{\partial x}&=\frac{\partial f}{\partial r}\frac{\partial r}{\partial x}+\frac{\partial f}{\partial \theta}\frac{\partial \theta}{\partial x}\\ &=\frac{\partial f}{\partial r}\frac{x}{r}-\frac{\partial f}{\partial \theta}\frac{y}{r^2} \end{align*} \]

\[ \frac{\partial f}{\partial y}=\frac{\partial f}{\partial r}\frac{y}{r}+\frac{\partial f}{\partial \theta}\frac{x}{r^2} \]

\[ \begin{align*} \therefore \nabla f&=\boldsymbol{e_r}(\frac{\partial f}{\partial x}\cos\theta+\frac{\partial f}{\partial y}\sin\theta)+\boldsymbol{e_\theta}(-\frac{\partial f}{\partial x}\sin\theta+\frac{\partial f}{\partial y}\cos\theta)+\boldsymbol{e_z}\frac{\partial f}{\partial z}\\ &=\boldsymbol{e_r}[\frac{x}{r}(f_r\frac{x}{r}-f_\theta\frac{y}{r^2})+\frac{y}{r}(f_r\frac{y}{r}+f_\theta\frac{x}{r^2})]\\ &+\boldsymbol{e_\theta}[-\frac{y}{r}(f_r\frac{x}{r}-f_\theta\frac{y}{r^2})+\frac{x}{r}(f_r\frac{y}{r}+f_\theta\frac{x}{r^2})]+\boldsymbol{e_z}f_z\\ &=\boldsymbol{e_r}\frac{\partial f}{\partial r}+\boldsymbol{e_\theta}\frac{1}{r}\frac{\partial f}{\partial \theta}+\boldsymbol{e_z}\frac{\partial f}{\partial z} \end{align*} \]

散度

\[ 柱坐标：\nabla \cdot \boldsymbol{A}=\frac{1}{r}\frac{\partial}{\partial r}(rA_r)+\frac{1}{r}\frac{\partial A_\theta}{\partial \theta}+\frac{\partial A_z}{\partial z} \]

\[ 球坐标：\nabla \cdot \boldsymbol{A}=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2A_r)+\frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}(\sin\theta A_\theta)+\frac{1}{r\sin\theta}\frac{\partial A_\phi}{\partial \phi} \]

以球坐标为例

我们已经拥有球坐标下的\(\nabla\)，接下来把它同一个球坐标点乘即可。

需要注意：在柱坐标和球坐标中，基向量会随空间位置的变化而改变，导致出现基向量的偏导不为0的情况。

\[ \begin{matrix} \displaystyle\frac{\partial \boldsymbol{e_r}}{\partial r}=0&\displaystyle\frac{\partial \boldsymbol{e_\theta}}{\partial r}=0&\displaystyle\frac{\partial \boldsymbol{e_\phi}}{\partial r}=0\\ \displaystyle\frac{\partial \boldsymbol{e_r}}{\partial \theta}=\boldsymbol{e_\theta}&\displaystyle\frac{\partial \boldsymbol{e_\theta}}{\partial \theta}=-\boldsymbol{e_r}&\displaystyle\frac{\partial \boldsymbol{e_\phi}}{\partial \theta}=0\\ \displaystyle\frac{\partial \boldsymbol{e_r}}{\partial \phi}=\sin\theta \boldsymbol{e_\phi}&\displaystyle\frac{\partial \boldsymbol{e_\theta}}{\partial\phi}=\cos\theta\boldsymbol{e_\phi}&\displaystyle\frac{\partial \boldsymbol{e_\phi}}{\partial \phi}=-\boldsymbol{e_r}\sin\theta-\boldsymbol{e_\theta}\cos\theta\\ \end{matrix} \]

\[ \begin{align*} \nabla\cdot\boldsymbol{A}&=\boldsymbol{e_r}\cdot\frac{\partial}{\partial r}(A_r\boldsymbol{e_r}+A_\theta\boldsymbol{e_\theta}+A_\phi\boldsymbol{e_\phi})\\ &+\frac{1}{r}\boldsymbol{e_\theta}\cdot\frac{\partial}{\partial \theta}(A_r\boldsymbol{e_r}+A_\theta\boldsymbol{e_\theta}+A_\phi\boldsymbol{e_\phi})\\ &+\frac{1}{r \sin\theta}\boldsymbol{e_\phi}\cdot\frac{\partial}{\partial \phi} (A_r\boldsymbol{e_r}+A_\theta\boldsymbol{e_\theta}+A_\phi\boldsymbol{e_\phi})\\ &=\boldsymbol{e_r}\cdot(\frac{\partial A_r}{\partial r}\boldsymbol{e_r}+\frac{\partial A_\theta}{\partial r}\boldsymbol{e_\theta}+\frac{\partial A_\phi}{\partial r}\boldsymbol{e_\phi})\\ &+\frac{1}{r}\boldsymbol{e_\theta}\cdot(\frac{\partial A_r}{\partial \theta}\boldsymbol{e_r}+\frac{\partial A_\theta}{\partial \theta}\boldsymbol{e_\theta}+\frac{\partial A_\phi}{\partial \phi}\boldsymbol{e_\phi}+A_r\boldsymbol{e_\theta}-A_\theta\boldsymbol{e_r})\\ &+\frac{1}{r\sin\theta}\boldsymbol{e_\phi}\cdot(\frac{\partial A_r}{\partial \phi}\boldsymbol{e_r}+\frac{\partial A_\theta}{\partial \phi}\boldsymbol{e_\theta}+\frac{\partial A_\phi}{\partial \phi}\boldsymbol{e_\phi}+A_r\sin\theta\boldsymbol{e_\phi}+A_\theta\cos\theta\boldsymbol{e_\phi}-(\sin\theta\boldsymbol{e_r}+\cos\theta\boldsymbol{e_\theta}))\\ &=\frac{\partial A_r}{\partial r}+\frac{1}{r}(\frac{\partial A_\theta}{\partial \theta}+A_r)+\frac{1}{r\sin\theta}(\frac{\partial A_\phi}{\partial \phi}+A_r\sin\theta+A_\theta\cos\theta)\\ &=\frac{1}{r^2}\frac{\partial }{\partial r}(r^2A_r)+\frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}(\sin\theta A_\theta)+\frac{1}{r\sin\theta}\frac{A_\phi}{\partial \phi} \end{align*}\]

旋度

\[ 柱坐标：\nabla\times \boldsymbol{A}=\frac{1}{r}\begin{vmatrix} \boldsymbol{e_r}&r\boldsymbol{e_\theta}&\boldsymbol{e_z}\\ \frac{\partial}{\partial r}&\frac{\partial}{\partial \theta}&\frac{\partial}{\partial z}\\ A_r&rA_\theta&A_z\\ \end{vmatrix} \]

\[ 球坐标：\nabla\times \boldsymbol{A}=\frac{1}{r^2\sin\theta}\begin{vmatrix} \boldsymbol{e_r}&r\boldsymbol{e_\theta}&r\sin\theta\boldsymbol{e_\phi}\\ \frac{\partial}{\partial r}&\frac{\partial}{\partial \theta}&\frac{\partial}{\partial \phi}\\ A_r&rA_\theta&r\sin\theta A_\phi\\ \end{vmatrix} \]

以球坐标为例

过程与散度推导类似。

\[ \begin{align*} \nabla\times\boldsymbol{A}&=\boldsymbol{e_r}\times(\frac{\partial A_r}{\partial r}\boldsymbol{e_r}+\frac{\partial A_\theta}{\partial r}\boldsymbol{e_\theta}+\frac{\partial A_\phi}{\partial r}\boldsymbol{e_\phi})\\ &+\frac{1}{r}\boldsymbol{e_\theta}\times(\frac{\partial A_r}{\partial \theta}\boldsymbol{e_r}+\frac{\partial A_\theta}{\partial \theta}\boldsymbol{e_\theta}+\frac{\partial A_\phi}{\partial \phi}\boldsymbol{e_\phi}+A_r\boldsymbol{e_\theta}-A_\theta\boldsymbol{e_r})\\ &+\frac{1}{r\sin\theta}\boldsymbol{e_\phi}\times(\frac{\partial A_r}{\partial \phi}\boldsymbol{e_r}+\frac{\partial A_\theta}{\partial \phi}\boldsymbol{e_\theta}+\frac{\partial A_\phi}{\partial \phi}\boldsymbol{e_\phi}+A_r\sin\theta\boldsymbol{e_\phi}+A_\theta\cos\theta\boldsymbol{e_\phi}-(\sin\theta\boldsymbol{e_r}+\cos\theta\boldsymbol{e_\theta}))\\ &=\boldsymbol{e_r}\frac{1}{r\sin\theta}(\frac{\partial}{\partial \theta}(\sin\theta A_\phi)-\frac{\partial A_\theta}{\partial \phi})+\boldsymbol{e_\theta}\frac{1}{r}(\frac{1}{\sin\theta}\frac{\partial A_r}{\partial \phi}-\frac{\partial}{\partial r}(rA_\phi))+\boldsymbol{e_\phi}\frac{1}{r}(\frac{\partial}{\partial r}(rA_\theta)-\frac{\partial A_r}{\partial \theta})\\ &=\frac{1}{r^2\sin\theta}\begin{vmatrix} \boldsymbol{e_r}&r\boldsymbol{e_\theta}&r\sin\theta\boldsymbol{e_\phi}\\ \frac{\partial}{\partial r}&\frac{\partial}{\partial \theta}&\frac{\partial}{\partial\phi}\\ A_r&rA_\theta&r\sin\theta A_\phi \end{vmatrix} \end{align*} \]