同一矢量和张量在不同坐标系下的转换

坐标系定义

球坐标系 $(R,\theta,\phi)$ ，直角坐标系 $(x, y, z)$

$x=R\sin\theta\cos\phi,\;y=R\sin\theta\cos\phi,\;z=R\cos\theta$

球坐标系和直角坐标系单位矢量转换

( $\hat{R},\hat{\theta},\hat{\phi}$ )为球坐标系的局部直角坐标单位矢量， $(\hat{x},\hat{y},\hat{z})$ 为全局坐标单位矢量

$\left(\begin{array}{c}\hat{R}\\\hat{\theta}\\\hat{\phi}\end{array}\right)=\left(\begin{array}{ccc}\sin\theta\cos\phi&\sin\theta\sin\phi&\cos\theta\\\cos\theta\cos\phi&\cos\theta\sin\phi&-\sin\theta\\-\sin\theta&\cos\varphi&0\end{array}\right)\left(\begin{array}{c}\hat{x}\\\hat{y}\\\hat{z}\end{array}\right)$

记 $\mathbf{A}=\left(\begin{array}{ccc}\sin\theta\cos\phi&\sin\theta\sin\phi&\cos\theta\\\cos\theta\cos\phi&\cos\theta\sin\phi&-\sin\theta\\-\sin\theta&\cos\varphi&0\end{array}\right)$ ， $\mathbf{A}$ 是正交矩阵， $\mathbf{A}^T\mathbf{A}=\mathbf{I}$

那么 $\left(\begin{array}{c}\hat{R}\\\hat{\theta}\\\hat{\phi}\end{array}\right)=\mathbf{A}\left(\begin{array}{c}\hat{x}\\\hat{y}\\\hat{z}\end{array}\right)$ ， $\left(\begin{array}{c}\hat{x}\\\hat{y}\\\hat{z}\end{array}\right)=\mathbf{A}^T\left(\begin{array}{c}\hat{R}\\\hat{\theta}\\\hat{\phi}\end{array}\right)$

不同坐标系的矢量转换

矢量 $\mathbf{g}=g_x\hat{x}+g_y\hat{y}+g_z\hat{z}=\left(\begin{array}{ccc}\hat{x}&\hat{y}&\hat{z}\end{array}\right)\left(\begin{array}{c}g_x\\g_y\\g_z\end{array}\right)=\left(\begin{array}{ccc}\hat{R}&\hat{\theta}&\hat{\phi}\end{array}\right)\mathbf{A}\left(\begin{array}{c}g_x\\g_y\\g_z\end{array}\right)$

所以 $\left(\begin{array}{c}g_R\\g_{\theta}\\g_{\phi}\end{array}\right)=\mathbf{A}\left(\begin{array}{c}g_{x}\\g_{y}\\g_{z}\end{array}\right)$

不同坐标系的张量转换

直角坐标系下的张量 $\mathbf{T}_{c}=\left(\begin{array}{ccc}T_{xx}&T_{xy}&T_{xz}\\T_{yx}&T_{yy}&T_{yz}\\T_{zx}&T_{zy}&T_{zz}\end{array}\right)$

写成分量形式

$\mathbf{T}=\begin{array}{c}T_{xx}\hat{x}\hat{x}+T_{xy}\hat{x}\hat{y}+T_{xz}\hat{x}\hat{z}\\+T_{yx}\hat{y}\hat{x}+T_{yy}\hat{y}\hat{y}+T_{yz}\hat{y}\hat{z}\\+T_{zx}\hat{z}\hat{x}+T_{zy}\hat{z}\hat{y}+T_{zz}\hat{z}\hat{z}\end{array}=\left(\begin{array}{ccc}\hat{x}&\hat{y}&\hat{z}\end{array}\right)\left(\begin{array}{ccc}T_{xx}&T_{xy}&T_{xz}\\T_{yx}&T_{yy}&T_{yz}\\T_{zx}&T_{zy}&T_{zz}\end{array}\right)\left(\begin{array}{c}\hat{x}\\\hat{y}\\\hat{z}\end{array}\right)$

代入 $\left(\begin{array}{c}\hat{x}\\\hat{y}\\\hat{z}\end{array}\right)=\mathbf{A}^T\left(\begin{array}{c}\hat{R}\\\hat{\theta}\\\hat{\phi}\end{array}\right)$ 得

$\mathbf{T}=\left(\begin{array}{ccc}\hat{R}&\hat{\theta}&\hat{\phi}\end{array}\right)\mathbf{A}\left(\begin{array}{ccc}T_{xx}&T_{xy}&T_{xz}\\T_{yx}&T_{yy}&T_{yz}\\T_{zx}&T_{zy}&T_{zz}\end{array}\right)\mathbf{A}^T\left(\begin{array}{c}\hat{R}\\\hat{\theta}\\\hat{\phi}\end{array}\right)$

所以，球坐标系下张量为

$\mathbf{T}_s=\mathbf{A}\mathbf{T}_c\mathbf{A}^T$

通过矢量的方向导数推导梯度张量的坐标系转换关系

$\hat{u}$ 是一个空间中的任意方向的单位矢量

矢量 $\mathbf{g}$ 沿着 $\hat{\mathbf{u}}$ 的方向导数为
$\mathbf{g}_u=(\hat{\mathbf{u}}\cdot \nabla)\mathbf{g}$

$\begin{aligned}(\hat{\mathbf{u}}\cdot \nabla)\mathbf{g}=&(u_x\frac{\partial}{\partial x}+u_y\frac{\partial}{\partial y}+u_z\frac{\partial}{\partial z})\mathbf{g}\\ =&(u_x\frac{\partial g_x}{\partial x}+u_y\frac{\partial g_x}{\partial y}+u_z\frac{\partial g_x}{\partial z})\hat{x}\\ &+(u_x\frac{\partial g_y}{\partial x}+u_y\frac{\partial g_y}{\partial y}+u_z\frac{\partial g_y}{\partial z})\hat{y}\\ &+(u_x\frac{\partial g_z}{\partial x}+u_y\frac{\partial g_z}{\partial y}+u_z\frac{\partial g_z}{\partial z})\hat{z}\\ =&\left(\begin{array}{ccc}T_{xx}&T_{xy}&T_{xz}\\T_{yx}&T_{yy}&T_{yz}\\T_{zx}&T_{zy}&T_{zz}\end{array}\right)\left(\begin{array}{c}u_{x}\\u_{y}\\u_{z}\end{array}\right)=\mathbf{T}\mathbf{u}\end{aligned}$

那么
$\mathbf{g}_u^{(s)}=\mathbf{A}\mathbf{g}_u^{(c)}$
上标 $(s)$ 表示球坐标， $(c)$ 表示直角坐标

又
$\mathbf{g}_u^{(c)}=\mathbf{T}_c\mathbf{\hat{u}}_c$
$\mathbf{\hat{u}}_c=\mathbf{A}^T\mathbf{\hat{u}}_s$

所以
$\mathbf{g}_u^{(s)}=\mathbf{A}\mathbf{T}_c\mathbf{A}^T\mathbf{\hat{u}}_s$

在球坐标系下也有
$\mathbf{g}_u^{(s)}=\mathbf{T}_s\mathbf{\hat{u}}_s$

所以
$\mathbf{T}_s=\mathbf{A}\mathbf{T}_c\mathbf{A}^T$