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,θ,ϕ)(R,\theta,\phi)(R,θ,ϕ)，直角坐标系(x,y,z)(x,y,z)(x,y,z)

x=Rsin⁡θcos⁡ϕ,y=Rsin⁡θcos⁡ϕ,z=Rcos⁡θx=R\sin\theta\cos\phi,\;y=R\sin\theta\cos\phi,\;z=R\cos\thetax=Rsinθcosϕ,y=Rsinθcosϕ,z=Rcosθ

球坐标系和直角坐标系单位矢量转换

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

(R^θ^ϕ^)=(sin⁡θcos⁡ϕsin⁡θsin⁡ϕcos⁡θcos⁡θcos⁡ϕcos⁡θsin⁡ϕ−sin⁡θ−sin⁡θcos⁡φ0)(x^y^z^)\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)⎝⎛R^θ^ϕ^⎠⎞=⎝⎛sinθcosϕcosθcosϕ−sinθsinθsinϕcosθsinϕcosφcosθ−sinθ0⎠⎞⎝⎛x^y^z^⎠⎞

记A=(sin⁡θcos⁡ϕsin⁡θsin⁡ϕcos⁡θcos⁡θcos⁡ϕcos⁡θsin⁡ϕ−sin⁡θ−sin⁡θcos⁡φ0)\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)A=⎝⎛sinθcosϕcosθcosϕ−sinθsinθsinϕcosθsinϕcosφcosθ−sinθ0⎠⎞，A\mathbf{A}A是正交矩阵，ATA=I\mathbf{A}^T\mathbf{A}=\mathbf{I}ATA=I

那么(R^θ^ϕ^)=A(x^y^z^)\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)⎝⎛R^θ^ϕ^⎠⎞=A⎝⎛x^y^z^⎠⎞，(x^y^z^)=AT(R^θ^ϕ^)\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)⎝⎛x^y^z^⎠⎞=AT⎝⎛R^θ^ϕ^⎠⎞

不同坐标系的矢量转换

矢量g=gxx^+gyy^+gzz^=(x^y^z^)(gxgygz)=(R^θ^ϕ^)A(gxgygz)\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)g=gxx^+gyy^+gzz^=(x^y^z^)⎝⎛gxgygz⎠⎞=(R^θ^ϕ^)A⎝⎛gxgygz⎠⎞

所以(gRgθgϕ)=A(gxgygz)\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)⎝⎛gRgθgϕ⎠⎞=A⎝⎛gxgygz⎠⎞

不同坐标系的张量转换

直角坐标系下的张量Tc=(TxxTxyTxzTyxTyyTyzTzxTzyTzz)\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)Tc=⎝⎛TxxTyxTzxTxyTyyTzyTxzTyzTzz⎠⎞

写成分量形式

T=Txxx^x^+Txyx^y^+Txzx^z^+Tyxy^x^+Tyyy^y^+Tyzy^z^+Tzxz^x^+Tzyz^y^+Tzzz^z^=(x^y^z^)(TxxTxyTxzTyxTyyTyzTzxTzyTzz)(x^y^z^)\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)T=Txxx^x^+Txyx^y^+Txzx^z^+Tyxy^x^+Tyyy^y^+Tyzy^z^+Tzxz^x^+Tzyz^y^+Tzzz^z^=(x^y^z^)⎝⎛TxxTyxTzxTxyTyyTzyTxzTyzTzz⎠⎞⎝⎛x^y^z^⎠⎞

代入(x^y^z^)=AT(R^θ^ϕ^)\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)⎝⎛x^y^z^⎠⎞=AT⎝⎛R^θ^ϕ^⎠⎞得

T=(R^θ^ϕ^)A(TxxTxyTxzTyxTyyTyzTzxTzyTzz)AT(R^θ^ϕ^)\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)T=(R^θ^ϕ^)A⎝⎛TxxTyxTzxTxyTyyTzyTxzTyzTzz⎠⎞AT⎝⎛R^θ^ϕ^⎠⎞

所以，球坐标系下张量为

Ts=ATcAT\mathbf{T}_s=\mathbf{A}\mathbf{T}_c\mathbf{A}^TTs=ATcAT

通过矢量的方向导数推导梯度张量的坐标系转换关系

u^\hat{u}u^是一个空间中的任意方向的单位矢量

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

(u^⋅∇)g=(ux∂∂x+uy∂∂y+uz∂∂z)g=(ux∂gx∂x+uy∂gx∂y+uz∂gx∂z)x^+(ux∂gy∂x+uy∂gy∂y+uz∂gy∂z)y^+(ux∂gz∂x+uy∂gz∂y+uz∂gz∂z)z^=(TxxTxyTxzTyxTyyTyzTzxTzyTzz)(uxuyuz)=Tu\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}(u^⋅∇)g===(ux∂x∂+uy∂y∂+uz∂z∂)g(ux∂x∂gx+uy∂y∂gx+uz∂z∂gx)x^+(ux∂x∂gy+uy∂y∂gy+uz∂z∂gy)y^+(ux∂x∂gz+uy∂y∂gz+uz∂z∂gz)z^⎝⎛TxxTyxTzxTxyTyyTzyTxzTyzTzz⎠⎞⎝⎛uxuyuz⎠⎞=Tu

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

又
gu(c)=Tcu^c\mathbf{g}_u^{(c)}=\mathbf{T}_c\mathbf{\hat{u}}_cgu(c)=Tcu^c
u^c=ATu^s\mathbf{\hat{u}}_c=\mathbf{A}^T\mathbf{\hat{u}}_su^c=ATu^s

所以
gu(s)=ATcATu^s\mathbf{g}_u^{(s)}=\mathbf{A}\mathbf{T}_c\mathbf{A}^T\mathbf{\hat{u}}_sgu(s)=ATcATu^s

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

所以
Ts=ATcAT\mathbf{T}_s=\mathbf{A}\mathbf{T}_c\mathbf{A}^TTs=ATcAT