Generalized Curvilinear Coordinate System
The Geometric Conservation Law - A link between finite-difference and finite-volume methods of flow computation on moving grids
Development of CFD Algorithms for Transient and Steady Aerodynamics- Fergal Boyle [Thesis]
two dimension static grid
\[\begin{split}\begin{array}{c}
x=x(\xi,\eta)=\phi_{1}(\xi,\eta)\\
y=y(\xi,\eta)=\phi_{2}(\xi,\eta)\\
\end{array}\end{split}\]
\[\begin{split}\begin{array}{c}
\xi=\xi(x,y)=\psi_{1}(x,y)\\
\eta=\eta(x,y)=\psi_{2}(x,y)\\
\end{array}\end{split}\]
\[\begin{split}\begin{array}{c}
\cfrac{\partial f}{\partial x}=\cfrac{\partial f}{\partial \xi}\cfrac{\partial \xi}{\partial x}+
\cfrac{\partial f}{\partial \eta}\cfrac{\partial \eta}{\partial x}\\
\cfrac{\partial f}{\partial y}=\cfrac{\partial f}{\partial \xi}\cfrac{\partial \xi}{\partial y}+
\cfrac{\partial f}{\partial \eta}\cfrac{\partial \eta}{\partial y}\\
\end{array}\end{split}\]
\[\begin{split}\begin{array}{c}
dx=x_{\xi}d\xi+x_{\eta}d\eta=({\phi_{1}})_{\xi}d\xi+({\phi_{1}})_{\eta}d\eta\\
dy=y_{\xi}d\xi+y_{\eta}d\eta=({\phi_{2}})_{\xi}d\xi+({\phi_{2}})_{\eta}d\eta\\
\end{array}\end{split}\]
\[\begin{split}\begin{array}{c}
\begin{bmatrix}
dx\\dy
\end{bmatrix}
=\begin{bmatrix}
x_{\xi}& x_{\eta}\\
y_{\xi}& y_{\eta}\\
\end{bmatrix}\begin{bmatrix}
d\xi\\d\eta
\end{bmatrix}
\end{array}\end{split}\]
\[\begin{split}\begin{array}{c}
d{\xi}={\xi}_{x}dx+{\xi}_{y}dy=(\psi_{1})_{x}dx+(\psi_{1})_{y}dy\\
d{\eta}={\eta}_{x}dx+{\eta}_{y}dy=(\psi_{2})_{x}dx+(\psi_{2})_{y}dy\\
\end{array}\end{split}\]
\[\begin{split}\begin{bmatrix}
d\xi\\d\eta
\end{bmatrix}
=\begin{bmatrix}
{\xi}_{x}& {\xi}_{y}\\
{\eta}_{x}& {\eta}_{y}\\
\end{bmatrix}\begin{bmatrix}
dx\\dy
\end{bmatrix}\end{split}\]
\[\begin{split}\begin{bmatrix}
d\xi\\d\eta
\end{bmatrix}
=\begin{bmatrix}
{\xi}_{x}& {\xi}_{y}\\
{\eta}_{x}& {\eta}_{y}\\
\end{bmatrix}\begin{bmatrix}
dx\\dy
\end{bmatrix}
= \begin{bmatrix}
{\xi}_{x}& {\xi}_{y}\\
{\eta}_{x}& {\eta}_{y}\\
\end{bmatrix}
\begin{bmatrix}
x_{\xi}& x_{\eta}\\
y_{\xi}& y_{\eta}\\
\end{bmatrix}\begin{bmatrix}
d\xi\\d\eta
\end{bmatrix}\end{split}\]
\[\begin{split}\begin{bmatrix}
{\xi}_{x}& {\xi}_{y}\\
{\eta}_{x}& {\eta}_{y}\\
\end{bmatrix}
\begin{bmatrix}
x_{\xi}& x_{\eta}\\
y_{\xi}& y_{\eta}\\
\end{bmatrix}=I\end{split}\]
\[\begin{split}\begin{bmatrix}
{\xi}_{x}& {\xi}_{y}\\
{\eta}_{x}& {\eta}_{y}\\
\end{bmatrix}=
\begin{bmatrix}
x_{\xi}& x_{\eta}\\
y_{\xi}& y_{\eta}\\
\end{bmatrix}^{-1}\end{split}\]
\[\begin{split}\begin{bmatrix}
x_{\xi}& x_{\eta}\\
y_{\xi}& y_{\eta}\\
\end{bmatrix}=\begin{bmatrix}
{\xi}_{x}& {\xi}_{y}\\
{\eta}_{x}& {\eta}_{y}\\
\end{bmatrix}^{-1}\end{split}\]
inverse of matrix
\[\mathbf{A}^{-1}=\cfrac{1}{\text{det}(\mathbf{A})}\text{adj}(\mathbf{A})=\cfrac{1}{|\mathbf{A}|}\mathbf{A}^{*}\]
For example,find the inverse matrix of the second-order matrix \(\mathbf{A}=\begin{pmatrix}a& b\\c&d\end{pmatrix}\)
\[|\mathbf{A}|=ad-bc\]
\[\begin{split}\mathbf{A}^{*}=\begin{pmatrix}d& -b\\-c&a\end{pmatrix}\\\end{split}\]
\[\begin{split}\mathbf{A}^{-1}=\cfrac{1}{|\mathbf{A}|}\mathbf{A}^{*} =\cfrac{1}{ad-bc}\begin{pmatrix}d& -b\\-c&a\end{pmatrix}\end{split}\]
Let
\[\begin{split}\mathbf{A}=\begin{bmatrix}x_{\xi}& x_{\eta}\\y_{\xi}& y_{\eta}\\\end{bmatrix}\end{split}\]
then
\[\begin{split}|\mathbf{A}|=x_{\xi}y_{\eta}-x_{\eta}y_{\xi}\\\end{split}\]
\[\begin{split}\mathbf{A}^{*}=\begin{pmatrix}y_{\eta}& -x_{\eta}\\-y_{\xi}&x_{\xi}\end{pmatrix}\\\end{split}\]
\[\begin{split}\mathbf{A}^{-1}=\cfrac{1}{|\mathbf{A}|}\mathbf{A}^{*} =\cfrac{1}{x_{\xi}y_{\eta}-x_{\eta}y_{\xi}}\begin{pmatrix}y_{\eta}& -x_{\eta}\\-y_{\xi}&x_{\xi}\end{pmatrix}\\\end{split}\]
Let
\[\begin{split}J=\begin{vmatrix}x_{\xi}& x_{\eta}\\y_{\xi}& y_{\eta}\\\end{vmatrix}\end{split}\]
then
\[\begin{split}\begin{bmatrix}{\xi}_{x}& {\xi}_{y}\\{\eta}_{x}& {\eta}_{y}\\\end{bmatrix} =\cfrac{1}{J}\begin{bmatrix}y_{\eta}& -x_{\eta}\\-y_{\xi}&x_{\xi}\end{bmatrix}\\\end{split}\]
\[\begin{split}\begin{array}{c}
{\xi}_{x}=\cfrac{1}{J}(y_{\eta})\quad{\xi}_{y}=\cfrac{1}{J}(-x_{\eta})\\
{\eta}_{x}=\cfrac{1}{J}(-y_{\xi})\quad{\eta}_{y}=\cfrac{1}{J}(x_{\xi})\\
\end{array}\end{split}\]
Equations in Cartesian Coordinates
\[\cfrac{\partial u}{\partial t}+\cfrac{\partial E}{\partial x}+\cfrac{\partial F}{\partial y}=0\]
\[\begin{split}\begin{array}{c}
\cfrac{\partial E}{\partial x}=\cfrac{\partial E}{\partial \xi}\cfrac{\partial \xi}{\partial x}+\cfrac{\partial E}{\partial \eta}\cfrac{\partial \eta}{\partial x}\\
\cfrac{\partial F}{\partial y}=\cfrac{\partial F}{\partial \xi}\cfrac{\partial \xi}{\partial y}+\cfrac{\partial F}{\partial \eta}\cfrac{\partial \eta}{\partial y}\\
\end{array}\end{split}\]
\[\cfrac{\partial u}{\partial t}+\cfrac{\partial E}{\partial \xi}\cfrac{\partial \xi}{\partial x}+\cfrac{\partial E}{\partial \eta}\cfrac{\partial \eta}{\partial x}
+\cfrac{\partial F}{\partial \xi}\cfrac{\partial \xi}{\partial y}+\cfrac{\partial F}{\partial \eta}\cfrac{\partial \eta}{\partial y}=0\]
\[\cfrac{\partial u}{\partial t}+\cfrac{\partial E}{\partial \xi}\cfrac{\partial \xi}{\partial x}+\cfrac{\partial F}{\partial \xi}\cfrac{\partial \xi}{\partial y}+\cfrac{\partial E}{\partial \eta}\cfrac{\partial \eta}{\partial x}
+\cfrac{\partial F}{\partial \eta}\cfrac{\partial \eta}{\partial y}=0\]
\[J\cfrac{\partial u}{\partial t}+J\cfrac{\partial E}{\partial \xi}\cfrac{\partial \xi}{\partial x}+J\cfrac{\partial F}{\partial \xi}\cfrac{\partial \xi}{\partial y}+J\cfrac{\partial E}{\partial \eta}\cfrac{\partial \eta}{\partial x}
+J\cfrac{\partial F}{\partial \eta}\cfrac{\partial \eta}{\partial y}=0\]
\[\cfrac{\partial (EJ\cfrac{\partial \xi}{\partial x})}{\partial \xi}=\cfrac{\partial E}{\partial \xi}(J\cfrac{\partial \xi}{\partial x})+E\cfrac{\partial (J\cfrac{\partial \xi}{\partial x})}{\partial \xi}\]
\[\begin{split}\cfrac{\partial (EJ\cfrac{\partial \eta}{\partial x})}{\partial \eta}=\cfrac{\partial E}{\partial \eta}(J\cfrac{\partial \eta}{\partial x})+E\cfrac{\partial (J\cfrac{\partial \eta}{\partial x})}{\partial \eta}\\\end{split}\]
\[\begin{split}\cfrac{\partial (FJ\cfrac{\partial \xi}{\partial y})}{\partial \xi}=\cfrac{\partial F}{\partial \xi}(J\cfrac{\partial \xi}{\partial y})+F\cfrac{\partial (J\cfrac{\partial \xi}{\partial y})}{\partial \xi}\\\end{split}\]
\[\begin{split}\cfrac{\partial (FJ\cfrac{\partial \eta}{\partial y})}{\partial \eta}=\cfrac{\partial F}{\partial \eta}(J\cfrac{\partial \eta}{\partial y})+F\cfrac{\partial (J\cfrac{\partial \eta}{\partial y})}{\partial \eta}\\\end{split}\]
\[E\cfrac{\partial (J\cfrac{\partial \xi}{\partial x})}{\partial \xi}
+E\cfrac{\partial (J\cfrac{\partial \eta}{\partial x})}{\partial \eta}
=E(\cfrac{\partial (y_{\eta})}{\partial \xi}+\cfrac{\partial (-y_{\xi})}{\partial \eta})=0\]
\[F\cfrac{\partial (J\cfrac{\partial \xi}{\partial y})}{\partial \xi}
+F\cfrac{\partial (J\cfrac{\partial \eta}{\partial y})}{\partial \eta}
=F(\cfrac{\partial (-x_{\eta})}{\partial \xi}+\cfrac{\partial (x_{\xi})}{\partial \eta})=0\]
\[J\cfrac{\partial u}{\partial t}
+\cfrac{\partial (EJ\cfrac{\partial \xi}{\partial x})}{\partial \xi}
+\cfrac{\partial (FJ\cfrac{\partial \xi}{\partial y})}{\partial \xi}
+\cfrac{\partial (EJ\cfrac{\partial \eta}{\partial x})}{\partial \eta}
+\cfrac{\partial (FJ\cfrac{\partial \eta}{\partial y})}{\partial \eta}=0\]
\[J\cfrac{\partial u}{\partial t}
+\cfrac{\partial (EJ\cfrac{\partial \xi}{\partial x})}{\partial \xi}
+\cfrac{\partial (FJ\cfrac{\partial \xi}{\partial y})}{\partial \xi}
+\cfrac{\partial (EJ\cfrac{\partial \eta}{\partial x})}{\partial \eta}
+\cfrac{\partial (FJ\cfrac{\partial \eta}{\partial y})}{\partial \eta}=0\]
\[J\cfrac{\partial u}{\partial t}
+\cfrac{\partial (J(\xi_{x}E+\xi_{y}F))}{\partial \xi}
+\cfrac{\partial (J(\eta_{x}E+\eta_{y}F))}{\partial \eta}
=0\]
two dimension dynamic grid
\[\begin{split}\begin{align}
x & = x(\xi,\eta,\tau) = \phi_{1}(\xi,\eta,\tau)\\
y & = y(\xi,\eta,\tau) = \phi_{2}(\xi,\eta,\tau)\\
t & = \tau
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\xi & = \xi(x,y,t) = \psi_{1}(x,y,t)\\
\eta & = \eta(x,y,t) = \psi_{2}(x,y,t)\\
\tau&=t
\end{align}\end{split}\]
\[\begin{split}J=\cfrac{\partial (x,y)}{\partial (\xi,\eta)}=\begin{vmatrix}x_{\xi}& x_{\eta}\\y_{\xi}& y_{\eta}\\\end{vmatrix}\end{split}\]
Equations in general curvilinear coordinate system
\[\begin{split}J(y_{1},y_{2},y_{3})=\cfrac{\partial (y_{1},y_{2},y_{3})}{\partial (\xi_{1},\xi_{2},\xi_{3})}
=\begin{vmatrix}
\cfrac{\partial y_{1}}{\partial \xi_{1}}&
\cfrac{\partial y_{1}}{\partial \xi_{2}}&
\cfrac{\partial y_{1}}{\partial \xi_{3}} \\
\cfrac{\partial y_{2}}{\partial \xi_{1}}&
\cfrac{\partial y_{2}}{\partial \xi_{2}}&
\cfrac{\partial y_{2}}{\partial \xi_{3}} \\
\cfrac{\partial y_{3}}{\partial \xi_{1}}&
\cfrac{\partial y_{3}}{\partial \xi_{2}}&
\cfrac{\partial y_{3}}{\partial \xi_{3}} \\
\end{vmatrix}\end{split}\]
\[\cfrac{\partial (y_{1},y_{2},\cdots,y_{i},y_{i+1},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{n})}
=-\cfrac{\partial (y_{1},y_{2},\cdots,y_{i+1},y_{i},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{n})}\]
\[\begin{split}\cfrac{\partial (y_{1},y_{2},\cdots,y_{i},y_{i+1},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{n})}
=\begin{vmatrix}
\cfrac{\partial y_{1}}{\partial \xi_{1}}& \cfrac{\partial y_{1}}{\partial \xi_{2}}&\cdots & \cfrac{\partial y_{1}}{\partial \xi_{n}}\\
\cfrac{\partial y_{2}}{\partial \xi_{1}}& \cfrac{\partial y_{2}}{\partial \xi_{2}}&\cdots & \cfrac{\partial y_{2}}{\partial \xi_{n}}\\
\vdots & \vdots&\ddots & \vdots\\
\cfrac{\partial y_{i}}{\partial \xi_{1}}& \cfrac{\partial y_{i}}{\partial \xi_{2}}&\cdots & \cfrac{\partial y_{i}}{\partial \xi_{n}}\\
\cfrac{\partial y_{i+1}}{\partial \xi_{1}}& \cfrac{\partial y_{i+1}}{\partial \xi_{2}}&\cdots & \cfrac{\partial y_{i+1}}{\partial \xi_{n}}\\
\vdots & \vdots&\ddots & \vdots\\
\cfrac{\partial y_{n}}{\partial \xi_{1}}& \cfrac{\partial y_{n}}{\partial \xi_{2}}&\cdots & \cfrac{\partial y_{n}}{\partial \xi_{n}}\\
\end{vmatrix}\end{split}\]
\[\begin{split}\cfrac{\partial (y_{1},y_{2},\cdots,y_{i+1},y_{i},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{n})}
=\begin{vmatrix}
\cfrac{\partial y_{1}}{\partial \xi_{1}}& \cfrac{\partial y_{1}}{\partial \xi_{2}}&\cdots & \cfrac{\partial y_{1}}{\partial \xi_{n}}\\
\cfrac{\partial y_{2}}{\partial \xi_{1}}& \cfrac{\partial y_{2}}{\partial \xi_{2}}&\cdots & \cfrac{\partial y_{2}}{\partial \xi_{n}}\\
\vdots & \vdots&\ddots & \vdots\\
\cfrac{\partial y_{i+1}}{\partial \xi_{1}}& \cfrac{\partial y_{i+1}}{\partial \xi_{2}}&\cdots & \cfrac{\partial y_{i+1}}{\partial \xi_{n}}\\
\cfrac{\partial y_{i}}{\partial \xi_{1}}& \cfrac{\partial y_{i}}{\partial \xi_{2}}&\cdots & \cfrac{\partial y_{i}}{\partial \xi_{n}}\\
\vdots & \vdots&\ddots & \vdots\\
\cfrac{\partial y_{n}}{\partial \xi_{1}}& \cfrac{\partial y_{n}}{\partial \xi_{2}}&\cdots & \cfrac{\partial y_{n}}{\partial \xi_{n}}\\
\end{vmatrix}\end{split}\]
\[\cfrac{\partial (y_{1},y_{2},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{i},\xi_{i+1},\cdots,\xi_{n})}
=-\cfrac{\partial (y_{1},y_{2},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{i+1},\xi_{i},\cdots,\xi_{n})}\]
\[\begin{split}\cfrac{\partial (y_{1},y_{2},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{i},\xi_{i+1},\cdots,\xi_{n})}
=\begin{vmatrix}
\cfrac{\partial y_{1}}{\partial \xi_{1}}&
\cfrac{\partial y_{1}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{1}}{\partial \xi_{i}}&
\cfrac{\partial y_{1}}{\partial \xi_{i+1}}&
\cdots&
\cfrac{\partial y_{1}}{\partial \xi_{n}}\\
\cfrac{\partial y_{2}}{\partial \xi_{1}}&
\cfrac{\partial y_{2}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{2}}{\partial \xi_{i}}&
\cfrac{\partial y_{2}}{\partial \xi_{i+1}}&
\cdots&
\cfrac{\partial y_{2}}{\partial \xi_{n}}\\
\vdots & \vdots&\ddots & \vdots& \vdots&\ddots& \vdots\\
\cfrac{\partial y_{n}}{\partial \xi_{1}}&
\cfrac{\partial y_{n}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{n}}{\partial \xi_{i}}&
\cfrac{\partial y_{n}}{\partial \xi_{i+1}}&
\cdots&
\cfrac{\partial y_{n}}{\partial \xi_{n}}\\
\end{vmatrix}\end{split}\]
\[\begin{split}\cfrac{\partial (y_{1},y_{2},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{i+1},\xi_{i},\cdots,\xi_{n})}
=\begin{vmatrix}
\cfrac{\partial y_{1}}{\partial \xi_{1}}&
\cfrac{\partial y_{1}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{1}}{\partial \xi_{i+1}}&
\cfrac{\partial y_{1}}{\partial \xi_{i}}&
\cdots&
\cfrac{\partial y_{1}}{\partial \xi_{n}}\\
\cfrac{\partial y_{2}}{\partial \xi_{1}}&
\cfrac{\partial y_{2}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{2}}{\partial \xi_{i+1}}&
\cfrac{\partial y_{2}}{\partial \xi_{i}}&
\cdots&
\cfrac{\partial y_{2}}{\partial \xi_{n}}\\
\vdots & \vdots&\ddots & \vdots& \vdots&\ddots& \vdots\\
\cfrac{\partial y_{n}}{\partial \xi_{1}}&
\cfrac{\partial y_{n}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{n}}{\partial \xi_{i+1}}&
\cfrac{\partial y_{n}}{\partial \xi_{i}}&
\cdots&
\cfrac{\partial y_{n}}{\partial \xi_{n}}\\
\end{vmatrix}\end{split}\]
\[\begin{split}\cfrac{\partial (y_{1},y_{2},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{n})}
=\begin{vmatrix}
\cfrac{\partial y_{1}}{\partial \xi_{1}}&
\cfrac{\partial y_{1}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{1}}{\partial \xi_{n}}\\
\cfrac{\partial y_{2}}{\partial \xi_{1}}&
\cfrac{\partial y_{2}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{2}}{\partial \xi_{n}}\\
\vdots & \vdots&\ddots & \vdots\\
\cfrac{\partial y_{n}}{\partial \xi_{1}}&
\cfrac{\partial y_{n}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{n}}{\partial \xi_{n}}\\
\end{vmatrix}\end{split}\]
When there are common variables between \(y_{i}\) and \(\xi_{i}\), then determinant reduction occurs, such as
\[\begin{split}\cfrac{\partial (\xi_{1},y_{2},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{n})}
=\begin{vmatrix}
\cfrac{\partial \xi_{1}}{\partial \xi_{1}}&
\cfrac{\partial \xi_{1}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial \xi_{1}}{\partial \xi_{n}}\\
\cfrac{\partial y_{2}}{\partial \xi_{1}}&
\cfrac{\partial y_{2}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{2}}{\partial \xi_{n}}\\
\vdots & \vdots&\ddots & \vdots\\
\cfrac{\partial y_{n}}{\partial \xi_{1}}&
\cfrac{\partial y_{n}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{n}}{\partial \xi_{n}}\\
\end{vmatrix}=
\begin{vmatrix}
1&
0&
\cdots&
0\\
\cfrac{\partial y_{2}}{\partial \xi_{1}}&
\cfrac{\partial y_{2}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{2}}{\partial \xi_{n}}\\
\vdots & \vdots&\ddots & \vdots\\
\cfrac{\partial y_{n}}{\partial \xi_{1}}&
\cfrac{\partial y_{n}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{n}}{\partial \xi_{n}}\\
\end{vmatrix}\end{split}\]
\[\begin{split}\cfrac{\partial (\xi_{1},y_{2},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{n})}
=\cfrac{\partial (y_{2},\cdots,y_{n})}{\partial (\xi_{2},\cdots,\xi_{n})}=
\begin{vmatrix}
\cfrac{\partial y_{2}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{2}}{\partial \xi_{n}}\\
\vdots & \ddots & \vdots\\
\cfrac{\partial y_{n}}{\partial \xi_{2}}&
\cdots&
\cfrac{\partial y_{n}}{\partial \xi_{n}}\\
\end{vmatrix}\end{split}\]
\[\cfrac{\partial (y_{1},y_{2},\cdots,y_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{n})}
=\cfrac{\partial (y_{1},y_{2},\cdots,y_{n})/\partial (\zeta_{1},\zeta_{2},\cdots,\zeta_{n})}{\partial (\xi_{1},\xi_{2},\cdots,\xi_{n})/\partial (\zeta_{1},\zeta_{2},\cdots,\zeta_{n})}=\cfrac{J(y_{1},y_{2},\cdots,y_{n})}{J(\xi_{1},\xi_{2},\cdots,\xi_{n})}\]
\[\cfrac{\partial U}{\partial t}+\cfrac{\partial E}{\partial x}+\cfrac{\partial F}{\partial y}=0\]
\[\begin{split}\begin{array}{c}
q_{0}=U,q_{1}=E,q_{2}=F\\
x_{0}=t,x_{1}=x,x_{2}=y\\
\xi_{0}=t,\xi_{1}=\xi,\xi_{2}=\eta\\
\end{array}\end{split}\]
\[\cfrac{\partial q_{i}}{\partial x_{i}}=0\]
\[Q_{k}=Jq_{i}\cfrac{\partial \xi_{k}}{\partial x_{i}}\]
\[\begin{split}\begin{align}
Q_{k} & = Jq_{0}\cfrac{\partial \xi_{k}}{\partial x_{0}}
+Jq_{1}\cfrac{\partial \xi_{k}}{\partial x_{1}}
+Jq_{2}\cfrac{\partial \xi_{k}}{\partial x_{2}}\\
& = Jq_{0}\cfrac{\partial \xi_{k}}{\partial t}
+Jq_{1}\cfrac{\partial \xi_{k}}{\partial x}
+Jq_{2}\cfrac{\partial \xi_{k}}{\partial y}
\end{align}\end{split}\]
\[\begin{align}
Q_{0}
& = Jq_{0}\cfrac{\partial t}{\partial t}
+Jq_{1}\cfrac{\partial t}{\partial x}
+Jq_{2}\cfrac{\partial t}{\partial y}=Jq_{0}
\end{align}\]
\[\begin{align}
Q_{1}
& = Jq_{0}\cfrac{\partial \xi}{\partial t}
+Jq_{1}\cfrac{\partial \xi}{\partial x}
+Jq_{2}\cfrac{\partial \xi}{\partial y}
\end{align}\]
\[\begin{align}
Q_{2}
& = Jq_{0}\cfrac{\partial \eta}{\partial t}
+Jq_{1}\cfrac{\partial \eta}{\partial x}
+Jq_{2}\cfrac{\partial \eta}{\partial y}
\end{align}\]
\[\cfrac{\partial U}{\partial t}
=\cfrac{\partial U}{\partial \tau}\cfrac{\partial \tau}{\partial t}
+\cfrac{\partial U}{\partial \xi}\cfrac{\partial \xi}{\partial t}
+\cfrac{\partial U}{\partial \eta}\cfrac{\partial \eta}{\partial t}\]
\[\cfrac{\partial U(x,y,t)}{\partial t}
=\cfrac{\partial \hat{U}(\xi,\eta,\tau)}{\partial \tau}\cfrac{\partial \tau}{\partial t}
+\cfrac{\partial \hat{U}(\xi,\eta,\tau)}{\partial \xi}\cfrac{\partial \xi}{\partial t}
+\cfrac{\partial \hat{U}(\xi,\eta,\tau)}{\partial \eta}\cfrac{\partial \eta}{\partial t}\]
\[\begin{align}
Q_{0}
& = JU\cfrac{\partial t}{\partial t}
+JE\cfrac{\partial t}{\partial x}
+JF\cfrac{\partial t}{\partial y}=JU
\end{align}\]
\[\begin{align}
Q_{1}
& = UJ\cfrac{\partial \xi}{\partial t}
+EJ\cfrac{\partial \xi}{\partial x}
+FJ\cfrac{\partial \xi}{\partial y}
\end{align}\]
\[\begin{align}
Q_{2}
& = UJ\cfrac{\partial \eta}{\partial t}
+EJ\cfrac{\partial \eta}{\partial x}
+FJ\cfrac{\partial \eta}{\partial y}
\end{align}\]
\[\cfrac{\partial (UJ\cfrac{\partial \xi}{\partial t})}{\partial \xi}=
\cfrac{\partial (U)}{\partial \xi}(J\cfrac{\partial \xi}{\partial t})+
(U)\cfrac{\partial }{\partial \xi}(J\cfrac{\partial \xi}{\partial t})\]
\[\begin{split}\begin{align}
x & = x(\xi,\eta,\tau) = \phi_{1}(\xi,\eta,\tau)\\
y & = y(\xi,\eta,\tau) = \phi_{2}(\xi,\eta,\tau)\\
t & = t(\xi,\eta,\tau)=\phi_{0}(\xi,\eta,\tau)=\tau\\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\xi & = \xi(x,y,t) = \psi_{1}(x,y,t)\\
\eta & = \eta(x,y,t) = \psi_{2}(x,y,t)\\
\tau&=\tau(x,y,t)= \psi_{0}(x,y,t)=t
\end{align}\end{split}\]