Generalized Curvilinear Coordinate System(3D)
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]
Applying the Curvilinear Transformation
The general curvilinear axes for a time-dependent curvilinear coordinate system are:
\[\begin{split}\begin{align}
\xi & = \xi(x,y,z,t)\\
\eta & = \eta(x,y,z,t)\\
\zeta & = \zeta (x,y,z,t)\\
\tau&=\tau(x,y,z,t)= t
\end{align}\end{split}\]
Using the chain rule for a function of multiple variables, the Cartesian derivatives can be written in terms of the curvilinear derivatives as:
\[\begin{split}\begin{align}
\cfrac{\partial }{\partial x}
& = \xi_{x} \cfrac{\partial}{\partial \xi}
+ \eta_{x} \cfrac{\partial}{\partial \eta}
+ \zeta_{x} \cfrac{\partial}{\partial \zeta}
+ \tau_{x} \cfrac{\partial}{\partial \tau}\\
\cfrac{\partial }{\partial y}
& = \xi_{y} \cfrac{\partial}{\partial \xi}
+ \eta_{y} \cfrac{\partial}{\partial \eta}
+ \zeta_{y} \cfrac{\partial}{\partial \zeta}
+ \tau_{y} \cfrac{\partial}{\partial \tau}\\
\cfrac{\partial }{\partial z}
& = \xi_{z} \cfrac{\partial}{\partial \xi}
+ \eta_{z} \cfrac{\partial}{\partial \eta}
+ \zeta_{z} \cfrac{\partial}{\partial \zeta}
+ \tau_{z} \cfrac{\partial}{\partial \tau}\\
\cfrac{\partial }{\partial t}
& = \xi_{t} \cfrac{\partial}{\partial \xi}
+ \eta_{t} \cfrac{\partial}{\partial \eta}
+ \zeta_{t} \cfrac{\partial}{\partial \zeta}
+ \tau_{t} \cfrac{\partial}{\partial \tau}
\end{align}\end{split}\]
where \(\xi_{x},\xi_{y},\xi_{z},\xi_{t},\eta_{x},\eta_{y},\eta_{z},\eta_{t},\zeta_{x},\zeta_{y},\zeta_{z},\zeta_{t}\) are the metrics of the transformation. (noting that \(\tau_{x}=\tau_{y}=\tau_{z}=0\) and \(\tau_{t}=1\)). The above equation can be written in the following matrix form:
\[\begin{split}\left[\begin{array}{c}
\frac{\partial}{\partial x} \\
\frac{\partial}{\partial y} \\
\frac{\partial}{\partial z} \\
\frac{\partial}{\partial t}
\end{array}\right]=\left[\begin{array}{llll}
\xi_{x} & \eta_{x} & \zeta_{x} & 0 \\
\xi_{y} & \eta_{y} & \zeta_{y} & 0 \\
\xi_{z} & \eta_{z} & \zeta_{z} & 0 \\
\xi_{t} & \eta_{t} & \zeta_{t} & 1
\end{array}\right]\left[\begin{array}{c}
\frac{\partial}{\partial \xi} \\
\frac{\partial}{\partial \eta} \\
\frac{\partial}{\partial \zeta} \\
\frac{\partial}{\partial \tau}
\end{array}\right]\end{split}\]
It is also possible to expand the curvilinear derivatives in terms of the Cartesian derivatives with the aid of the chain rule:
\[\begin{split}\begin{aligned}
\frac{\partial}{\partial \xi} & =x_{\xi} \frac{\partial}{\partial x}+y_{\xi} \frac{\partial}{\partial y}+z_{\xi} \frac{\partial}{\partial z}+t_{\xi} \frac{\partial}{\partial t} \\
\frac{\partial}{\partial \eta} & =x_{\eta} \frac{\partial}{\partial x}+y_{\eta} \frac{\partial}{\partial y}+z_{\eta} \frac{\partial}{\partial z}+t_{\eta} \frac{\partial}{\partial t} \\
\frac{\partial}{\partial \zeta} & =x_{\zeta} \frac{\partial}{\partial x}+y_{\zeta} \frac{\partial}{\partial y}+z_{\zeta} \frac{\partial}{\partial z}+t_{\zeta} \frac{\partial}{\partial t} \\
\frac{\partial}{\partial \tau} & =x_{\tau} \frac{\partial}{\partial x}+y_{\tau} \frac{\partial}{\partial y}+z_{\tau} \frac{\partial}{\partial z}+t_{\tau} \frac{\partial}{\partial t}
\end{aligned}\end{split}\]
Note again that \(t_{\xi}=t_{\eta}=t_{\zeta}=0\) and \(t_{\tau}=1\). The equation set above can also be written in matrix form:
\[\begin{split}\left[\begin{array}{c}
\frac{\partial}{\partial \xi} \\
\frac{\partial}{\partial \eta} \\
\frac{\partial}{\partial \zeta} \\
\frac{\partial}{\partial \tau}
\end{array}\right]
=\left[\begin{array}{llll}
{x}_{\xi} & {y}_{\xi}& {z}_{\xi} & 0 \\
{x}_{\eta} & {y}_{\eta}& {z}_{\eta} & 0 \\
{x}_{\zeta} & {y}_{\zeta}& {z}_{\zeta} & 0 \\
x_{\tau} & y_{\tau} & z_{\tau} & 1
\end{array}\right]=\left[\begin{array}{c}
\frac{\partial}{\partial x} \\
\frac{\partial}{\partial y} \\
\frac{\partial}{\partial z} \\
\frac{\partial}{\partial t}
\end{array}\right]\end{split}\]
Comparing the above equation sets, it is evident that the following holds true:
\[\begin{split}\left[\begin{array}{llll}
\xi_{x} & \eta_{x} & \zeta_{x} & 0 \\
\xi_{y} & \eta_{y} & \zeta_{y} & 0 \\
\xi_{z} & \eta_{z} & \zeta_{z} & 0 \\
\xi_{t} & \eta_{t} & \zeta_{t} & 1
\end{array}\right]=
\left[\begin{array}{llll}
{x}_{\xi} & {y}_{\xi}& {z}_{\xi} & 0 \\
{x}_{\eta} & {y}_{\eta}& {z}_{\eta} & 0 \\
{x}_{\zeta} & {y}_{\zeta}& {z}_{\zeta} & 0 \\
x_{\tau} & y_{\tau} & z_{\tau} & 1
\end{array}\right]^{-1}\end{split}\]
If the 4x4 matrix on the right-hand side is inverted, it is then possible to solve for the metrics of the transformation. The inverse of a matrix \(\mathbf{D}\) can be obtained using the following expression:
\[\mathbf{D}=\frac{1}{determinant(\mathbf{D})}adjoint(\mathbf{D})\]
Let
\[\begin{split}A=\left[\begin{array}{llll}
{x}_{\xi} & {y}_{\xi}& {z}_{\xi} & 0 \\
{x}_{\eta} & {y}_{\eta}& {z}_{\eta} & 0 \\
{x}_{\zeta} & {y}_{\zeta}& {z}_{\zeta} & 0 \\
x_{\tau} & y_{\tau} & z_{\tau} & 1
\end{array}\right]\end{split}\]
The determinant of the matrix \(A\) is:
\[\begin{split}\text{det}{A}=
\left|\begin{array}{llll}
{x}_{\xi} & {y}_{\xi}& {z}_{\xi} & 0 \\
{x}_{\eta} & {y}_{\eta}& {z}_{\eta} & 0 \\
{x}_{\zeta} & {y}_{\zeta}& {z}_{\zeta} & 0 \\
x_{\tau} & y_{\tau} & z_{\tau} & 1
\end{array}\right|
={x}_{\xi}({y}_{\eta}{z}_{\zeta}-{z}_{\eta}{y}_{\zeta})
-{y}_{\xi}({x}_{\eta}{z}_{\zeta}-{z}_{\eta}{x}_{\zeta})
+{z}_{\xi}({x}_{\eta}{y}_{\zeta}-{y}_{\eta}{x}_{\zeta})\end{split}\]
The Jacobian of the inverse transformation is defined as:
\[\begin{split}J^{-1}=\cfrac{\partial (x,y,z,t)}{\partial (\xi,\eta,\zeta,\tau)}=
\left|\begin{array}{llll}
{x}_{\xi} & {y}_{\xi}& {z}_{\xi} & 0 \\
{x}_{\eta} & {y}_{\eta}& {z}_{\eta} & 0 \\
{x}_{\zeta} & {y}_{\zeta}& {z}_{\zeta} & 0 \\
x_{\tau} & y_{\tau} & z_{\tau} & 1
\end{array}\right|\end{split}\]
\[J^{-1}=\text{det}({A})\]
Following the matrix inversion the metrics of the transformation are evaluated as:
\[A^{-1}=\cfrac{1}{\text{det}{A}}A^{*}\]
\[\begin{split}\mathbf{A}^{*}=\begin{bmatrix}
A_{11}&A_{21} &\cdots & A_{n1}\\
A_{12}&A_{22} &\cdots & A_{n2}\\
\vdots& \vdots & &\vdots \\
A_{1n}&A_{2n} &\cdots & A_{nn}\\
\end{bmatrix}\end{split}\]
\[A_{ij}=(-1)^{i+j}M_{ij}\]
then
\[\begin{split}A^{-1}=J\begin{bmatrix}
A_{11}& A_{21} & A_{31} & A_{41}\\
A_{12}& A_{22} & A_{32} & A_{42}\\
A_{13}& A_{23} & A_{33} & A_{43}\\
A_{14}& A_{24} & A_{34} & A_{44}\\
\end{bmatrix}\end{split}\]
\[\begin{split}\left[\begin{array}{llll}
\xi_{x} & \eta_{x} & \zeta_{x} & 0 \\
\xi_{y} & \eta_{y} & \zeta_{y} & 0 \\
\xi_{z} & \eta_{z} & \zeta_{z} & 0 \\
\xi_{t} & \eta_{t} & \zeta_{t} & 1
\end{array}\right]=
A^{-1}=J\begin{bmatrix}
A_{11}& A_{21} & A_{31} & A_{41}\\
A_{12}& A_{22} & A_{32} & A_{42}\\
A_{13}& A_{23} & A_{33} & A_{43}\\
A_{14}& A_{24} & A_{34} & A_{44}\\
\end{bmatrix}\end{split}\]
specifically:
\[\begin{split}A_{11}=
+\left|\begin{array}{llll}
{y}_{\eta}& {z}_{\eta} & 0 \\
{y}_{\zeta}& {z}_{\zeta} & 0 \\
y_{\tau} & z_{\tau} & 1
\end{array}\right| =+({y}_{\eta}{z}_{\zeta}-{z}_{\eta}{y}_{\zeta})\end{split}\]
\[\begin{split}A_{12}=
-\left|\begin{array}{llll}
{x}_{\eta} & {z}_{\eta} & 0 \\
{x}_{\zeta} & {z}_{\zeta} & 0 \\
x_{\tau} & z_{\tau} & 1
\end{array}\right| =-({x}_{\eta}{z}_{\zeta}-{z}_{\eta}{x}_{\zeta})\end{split}\]
\[\begin{split}A_{13}
=+ \left|\begin{array}{llll}
{x}_{\eta} & {y}_{\eta} & 0 \\
{x}_{\zeta} & {y}_{\zeta} & 0 \\
x_{\tau} & y_{\tau} & 1
\end{array}\right| =+({x}_{\eta}{y}_{\zeta}-{y}_{\eta}{x}_{\zeta}) \\\end{split}\]
\[\begin{split}\begin{align}
A_{14} & = - \left|\begin{array}{lll}
{x}_{\eta} & {y}_{\eta}& {z}_{\eta}\\
{x}_{\zeta} & {y}_{\zeta}& {z}_{\zeta} \\
x_{\tau} & y_{\tau} & z_{\tau}
\end{array}\right| = -x_{\tau}\left|\begin{array}{ll}
{y}_{\eta}& {z}_{\eta}\\
{y}_{\zeta}& {z}_{\zeta} \\
\end{array}\right|
+y_{\tau}\left|\begin{array}{ll}
{x}_{\eta} & {z}_{\eta}\\
{x}_{\zeta} & {z}_{\zeta} \\
\end{array}\right|
-z_{\tau}\left|\begin{array}{ll}
{x}_{\eta} & {y}_{\eta}\\
{x}_{\zeta} & {y}_{\zeta} \\
\end{array}\right|\\
&=-x_{\tau}({y}_{\eta}{z}_{\zeta}-{y}_{\zeta}{z}_{\eta})
-y_{\tau}({z}_{\eta}{x}_{\zeta}-{z}_{\zeta}{x}_{\eta})
-z_{\tau}({x}_{\eta}{y}_{\zeta}-{x}_{\zeta}{y}_{\eta})
\end{align}\end{split}\]
\[\begin{split}A_{21}
=-\left|\begin{array}{llll}
{y}_{\xi}& {z}_{\xi} & 0 \\
{y}_{\zeta}& {z}_{\zeta} & 0 \\
y_{\tau} & z_{\tau} & 1
\end{array}\right| =-({y}_{\xi}{z}_{\zeta}-{z}_{\xi}{y}_{\zeta})\end{split}\]
\[\begin{split}A_{22}
= +\left|\begin{array}{llll}
{x}_{\xi} & {z}_{\xi} & 0 \\
{x}_{\zeta} & {z}_{\zeta} & 0 \\
x_{\tau} & z_{\tau} & 1
\end{array}\right| =({x}_{\xi}{z}_{\zeta}-{z}_{\xi}{x}_{\zeta})\\\end{split}\]
\[\begin{split}A_{23}
=- \left|\begin{array}{llll}
{x}_{\xi} & {y}_{\xi} & 0 \\
{x}_{\zeta} & {y}_{\zeta} & 0 \\
x_{\tau} & y_{\tau} & 1
\end{array}\right|=- ({x}_{\xi}{y}_{\zeta}-{y}_{\xi}{x}_{\zeta})\\\end{split}\]
\[\begin{split}\begin{align}
A_{24} & = + \left|\begin{array}{llll}
{x}_{\xi} & {y}_{\xi}& {z}_{\xi}\\
{x}_{\zeta} & {y}_{\zeta}& {z}_{\zeta} \\
x_{\tau} & y_{\tau} & z_{\tau}
\end{array}\right| = x_{\tau}\left|\begin{array}{llll}
{y}_{\xi}& {z}_{\xi}\\
{y}_{\zeta}& {z}_{\zeta} \\
\end{array}\right|
-y_{\tau}\left|\begin{array}{llll}
{x}_{\xi} & {z}_{\xi}\\
{x}_{\zeta} & {z}_{\zeta} \\
\end{array}\right|
+z_{\tau}\left|\begin{array}{llll}
{x}_{\xi} & {y}_{\xi}\\
{x}_{\zeta} & {y}_{\zeta} \\
\end{array}\right|\\
&=x_{\tau}({y}_{\xi}{z}_{\zeta}-{z}_{\xi}{y}_{\zeta})
-y_{\tau}({x}_{\xi}{z}_{\zeta}-{z}_{\xi}{x}_{\zeta})
+z_{\tau}({x}_{\xi}{y}_{\zeta}-{y}_{\xi}{x}_{\zeta})
\end{align}\end{split}\]
\[\begin{split}A_{31}
= + \left|\begin{array}{llll}
{y}_{\xi}& {z}_{\xi} & 0 \\
{y}_{\eta}& {z}_{\eta} & 0 \\
y_{\tau} & z_{\tau} & 1
\end{array}\right|=+({y}_{\xi}{z}_{\eta}-{z}_{\xi}{y}_{\eta})\end{split}\]
\[\begin{split}A_{32}
=- \left|\begin{array}{llll}
{x}_{\xi} & {z}_{\xi} & 0 \\
{x}_{\eta} & {z}_{\eta} & 0 \\
x_{\tau} & z_{\tau} & 1
\end{array}\right| =-({x}_{\xi}{z}_{\eta}-{z}_{\xi}{x}_{\eta})\\\end{split}\]
\[\begin{split}A_{33}
= +\left|\begin{array}{llll}
{x}_{\xi} & {y}_{\xi} & 0 \\
{x}_{\eta} & {y}_{\eta} & 0 \\
x_{\tau} & y_{\tau} & 1
\end{array}\right| =+({x}_{\xi}{y}_{\eta}-{y}_{\xi}{x}_{\eta})\\\end{split}\]
\[\begin{split}\begin{align}
A_{34}
&=-\left|\begin{array}{llll}
{x}_{\xi} & {y}_{\xi}& {z}_{\xi} \\
{x}_{\eta} & {y}_{\eta}& {z}_{\eta} \\
x_{\tau} & y_{\tau} & z_{\tau}
\end{array}\right|
=-x_{\tau}\left|\begin{array}{llll}
{y}_{\xi}& {z}_{\xi} \\
{y}_{\eta}& {z}_{\eta} \\
\end{array}\right|
+y_{\tau}\left|\begin{array}{llll}
{x}_{\xi} & {z}_{\xi} \\
{x}_{\eta} & {z}_{\eta} \\
\end{array}\right|
-z_{\tau}\left|\begin{array}{llll}
{x}_{\xi} & {y}_{\xi} \\
{x}_{\eta} & {y}_{\eta} \\
\end{array}\right|\\
&=-x_{\tau}({y}_{\xi}{z}_{\eta}-{z}_{\xi}{y}_{\eta})
+y_{\tau}({x}_{\xi}{z}_{\eta}-{z}_{\xi}{x}_{\eta})
-z_{\tau}({x}_{\xi}{y}_{\eta}-{y}_{\xi}{x}_{\eta})\\
\end{align}\end{split}\]
\[\begin{split}A_{41}
=-\left|\begin{array}{llll}
{y}_{\xi}& {z}_{\xi} & 0 \\
{y}_{\eta}& {z}_{\eta} & 0 \\
{y}_{\zeta}& {z}_{\zeta} & 0 \\
\end{array}\right| =0\end{split}\]
\[\begin{split}A_{42}
=+\left|\begin{array}{llll}
{x}_{\xi} & {z}_{\xi} & 0 \\
{x}_{\eta} & {z}_{\eta} & 0 \\
{x}_{\zeta} & {z}_{\zeta} & 0 \\
\end{array}\right|=0\end{split}\]
\[\begin{split}A_{43}
=-\left|\begin{array}{llll}
{x}_{\xi} & {y}_{\xi} & 0 \\
{x}_{\eta} & {y}_{\eta} & 0 \\
{x}_{\zeta} & {y}_{\zeta} & 0 \\
\end{array}\right|=0\end{split}\]
\[\begin{split}A_{44}
=+ \left|\begin{array}{llll}
{x}_{\xi} & {y}_{\xi}& {z}_{\xi}\\
{x}_{\eta} & {y}_{\eta}& {z}_{\eta} \\
{x}_{\zeta} & {y}_{\zeta}& {z}_{\zeta} \\
\end{array}\right|=J^{-1}\end{split}\]
Finally:
\[\begin{split}\begin{align}
\xi_{x} = JA_{11} & = +J({y}_{\eta}{z}_{\zeta}-{z}_{\eta}{y}_{\zeta})\\
\xi_{y} = JA_{12} & = -J({x}_{\eta}{z}_{\zeta}-{z}_{\eta}{x}_{\zeta})\\
\xi_{z} = JA_{13} & = +J({x}_{\eta}{y}_{\zeta}-{y}_{\eta}{x}_{\zeta})\\
\xi_{t} = JA_{14} & = +J(-x_{\tau}({y}_{\eta}{z}_{\zeta}-{y}_{\zeta}{z}_{\eta})
-y_{\tau}({z}_{\eta}{x}_{\zeta}-{z}_{\zeta}{x}_{\eta})
-z_{\tau}({x}_{\eta}{y}_{\zeta}-{x}_{\zeta}{y}_{\eta}))\\
& = -x_{\tau}\xi_{x}-y_{\tau}\xi_{y}-z_{\tau}\xi_{z}
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\eta_{x} = JA_{21} & = -J({y}_{\xi}{z}_{\zeta}-{z}_{\xi}{y}_{\zeta})\\
\eta_{y} = JA_{22} & = +J({x}_{\xi}{z}_{\zeta}-{z}_{\xi}{x}_{\zeta})\\
\eta_{z} = JA_{23} & = -J({x}_{\xi}{y}_{\zeta}-{y}_{\xi}{x}_{\zeta})\\
\eta_{t} = JA_{24} & = +J(x_{\tau}({y}_{\xi}{z}_{\zeta}-{z}_{\xi}{y}_{\zeta})
-y_{\tau}({x}_{\xi}{z}_{\zeta}-{z}_{\xi}{x}_{\zeta})
+z_{\tau}({x}_{\xi}{y}_{\zeta}-{y}_{\xi}{x}_{\zeta}))\\
& = -x_{\tau}\eta_{x}-y_{\tau}\eta_{y}-z_{\tau}\eta_{z}
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\zeta_{x} = JA_{31} & = +J({y}_{\xi}{z}_{\eta}-{z}_{\xi}{y}_{\eta})\\
\zeta_{y} = JA_{32} & = -J({x}_{\xi}{z}_{\eta}-{z}_{\xi}{x}_{\eta})\\
\zeta_{z} = JA_{33} & = +J({x}_{\xi}{y}_{\eta}-{y}_{\xi}{x}_{\eta})\\
\zeta_{t} = JA_{34} & = +J(-x_{\tau}({y}_{\xi}{z}_{\eta}-{z}_{\xi}{y}_{\eta})
+y_{\tau}({x}_{\xi}{z}_{\eta}-{z}_{\xi}{x}_{\eta})
-z_{\tau}({x}_{\xi}{y}_{\eta}-{y}_{\xi}{x}_{\eta}))\\
& = -x_{\tau}\zeta_{x}-y_{\tau}\zeta_{y}-z_{\tau}\zeta_{z}
\end{align}\end{split}\]
Euler Equations in Cartesian Coordinates
The partial differential equation form of the non-dimensional, three-dimensional, Euler equations in Cartesian coordinates in an inertial reference frame, neglecting volumetric heat addition and body forces, is:
\[\cfrac{\partial \mathbf{q}}{\partial \text{t}}+
\cfrac{\partial \mathbf{f}}{\partial \text{x}}+
\cfrac{\partial \mathbf{g}}{\partial \text{y}}+
\cfrac{\partial \mathbf{h}}{\partial \text{z}}=0\]
where the vector of sonserved variables, \(\mathbf{q}\) , and the vectors of the inviscid flux terms, \(\mathbf{f}\), \(\mathbf{g}\), and \(\mathbf{h}\), are:
\[\begin{split}\begin{array}{l}
\mathbf{q}=\begin{bmatrix}
\rho\\ \rho u\\ \rho v\\ \rho w \\ \rho E\\
\end{bmatrix} \quad
\mathbf{f}=\begin{bmatrix}
\rho u\\ \rho uu+p\\ \rho vu\\ \rho wu \\ \rho Hu\\
\end{bmatrix} \quad
\mathbf{g}=\begin{bmatrix}
\rho v\\ \rho uv\\ \rho vv+p\\ \rho wv \\ \rho Hv\\
\end{bmatrix} \quad
\mathbf{h}=\begin{bmatrix}
\rho w\\ \rho uw\\ \rho vw\\ \rho ww+p \\ \rho Hw\\
\end{bmatrix} \quad
\end{array}\end{split}\]
where \(\rho\) is the density, \(p\) is the static pressure, \(u\), \(v\) and \(w\) are the Cartesian velocity components in the \(x\), \(y\) and \(z\) directions respectively, \(E\) is the total energy perunit mass and \(H\) is the total enthalpy per unit mass.
\[\cfrac{\partial\mathbf{q} }{\partial \text{t}}
= \xi_{t} \cfrac{\partial \mathbf{\hat{q}}}{\partial \xi}
+ \eta_{t} \cfrac{\partial\mathbf{\hat{q}}}{\partial \eta}
+ \zeta_{t} \cfrac{\partial\mathbf{\hat{q}}}{\partial \zeta}
+ \tau_{t} \cfrac{\partial\mathbf{\hat{q}}}{\partial \tau}\]
\[\begin{split}\cfrac{\partial \mathbf{f}}{\partial \text{x}}
& = \xi_{x} \cfrac{\partial\mathbf{\hat{f}}}{\partial \xi}
+ \eta_{x} \cfrac{\partial\mathbf{\hat{f}}}{\partial \eta}
+ \zeta_{x} \cfrac{\partial\mathbf{\hat{f}}}{\partial \zeta}
+ \tau_{x} \cfrac{\partial\mathbf{\hat{f}}}{\partial \tau}\\\end{split}\]
\[\begin{split}\cfrac{\partial \mathbf{g}}{\partial \text{y}}
= \xi_{y} \cfrac{\partial\mathbf{\hat{g}}}{\partial \xi}
+ \eta_{y} \cfrac{\partial\mathbf{\hat{g}}}{\partial \eta}
+ \zeta_{y} \cfrac{\partial\mathbf{\hat{g}}}{\partial \zeta}
+ \tau_{y} \cfrac{\partial\mathbf{\hat{g}}}{\partial \tau}\\\end{split}\]
\[\begin{split}\cfrac{\partial \mathbf{h}}{\partial \text{z}}
= \xi_{z} \cfrac{\partial\mathbf{\hat{h}}}{\partial \xi}
+ \eta_{z} \cfrac{\partial\mathbf{\hat{h}}}{\partial \eta}
+ \zeta_{z} \cfrac{\partial\mathbf{\hat{h}}}{\partial \zeta}
+ \tau_{z} \cfrac{\partial\mathbf{\hat{h}}}{\partial \tau}\\\end{split}\]
multiply by \(J^{-1}\) to get:
\[\begin{split}J^{-1}(\cfrac{\partial \mathbf{q}}{\partial \text{t}}+
\cfrac{\partial \mathbf{f}}{\partial \text{x}}+
\cfrac{\partial \mathbf{g}}{\partial \text{y}}+
\cfrac{\partial \mathbf{h}}{\partial \text{z}})=0\\\end{split}\]
\[\begin{split}\begin{align}
0 & = \hat\xi_{t} \cfrac{\partial \mathbf{\hat{q}}}{\partial \xi}
+ \hat\eta_{t} \cfrac{\partial\mathbf{\hat{q}}}{\partial \eta}
+ \hat\zeta_{t} \cfrac{\partial\mathbf{\hat{q}}}{\partial \zeta}
+ \hat\tau_{t} \cfrac{\partial\mathbf{\hat{q}}}{\partial \tau}\\
&+ \hat\xi_{x} \cfrac{\partial\mathbf{\hat{f}}}{\partial \xi}
+ \hat\eta_{x} \cfrac{\partial\mathbf{\hat{f}}}{\partial \eta}
+ \hat\zeta_{x} \cfrac{\partial\mathbf{\hat{f}}}{\partial \zeta}
+ \hat\tau_{x} \cfrac{\partial\mathbf{\hat{f}}}{\partial \tau}\\
&+ \hat\xi_{y} \cfrac{\partial\mathbf{\hat{g}}}{\partial \xi}
+ \hat\eta_{y} \cfrac{\partial\mathbf{\hat{g}}}{\partial \eta}
+ \hat\zeta_{y} \cfrac{\partial\mathbf{\hat{g}}}{\partial \zeta}
+ \hat\tau_{y} \cfrac{\partial\mathbf{\hat{g}}}{\partial \tau}\\
&+ \hat\xi_{z} \cfrac{\partial\mathbf{\hat{h}}}{\partial \xi}
+ \hat\eta_{z} \cfrac{\partial\mathbf{\hat{h}}}{\partial \eta}
+ \hat\zeta_{z} \cfrac{\partial\mathbf{\hat{h}}}{\partial \zeta}
+ \hat\tau_{z} \cfrac{\partial\mathbf{\hat{h}}}{\partial \tau}\\
\end{align}\end{split}\]
where
\[\begin{split}\begin{align}
(\hat\xi_{x},\hat\xi_{y},\hat\xi_{z},\hat\xi_{t}) & = J^{-1}(\xi_{x},\xi_{y},\xi_{z},\xi_{t})\\
(\hat\eta_{x},\hat\eta_{y},\hat\eta_{z},\hat\eta_{t}) & = J^{-1}(\eta_{x},\eta_{y},\eta_{z},\eta_{t})\\
(\hat\zeta_{x},\hat\zeta_{y},\hat\zeta_{z},\hat\zeta_{t}) & = J^{-1}(\zeta_{x},\zeta_{y},\zeta_{z},\zeta_{t})\\
(\hat\tau_{x},\hat\tau_{y},\hat\tau_{z},\hat\tau_{t}) & = J^{-1}(\tau_{x},\tau_{y},\tau_{z},\tau_{t}) = J^{-1}(0,0,0,1)\\
\end{align}\end{split}\]
then
\[\begin{split}\begin{align}
0 & = \cfrac{\partial (\hat\xi_{t}\mathbf{\hat{q}})}{\partial \xi}
+ \cfrac{\partial(\hat\eta_{t}\mathbf{\hat{q}})}{\partial \eta}
+\cfrac{\partial(\hat\zeta_{t}\mathbf{\hat{q}})}{\partial \zeta}
+ \cfrac{\partial(\hat\tau_{t}\mathbf{\hat{q}})}{\partial \tau}\\
&-\mathbf{\hat{q}}
(\cfrac{\partial (\hat\tau_{t}\equiv J^{-1})}{\partial \tau}
+\cfrac{\partial (\hat\xi_{t})}{\partial \xi}
+\cfrac{\partial(\hat\eta_{t})}{\partial \eta}
+\cfrac{\partial(\hat\zeta_{t})}{\partial \zeta}
)\\
&+ \cfrac{\partial(\hat\xi_{x}\mathbf{\hat{f}})}{\partial \xi}
+ \cfrac{\partial(\hat\eta_{x}\mathbf{\hat{f}})}{\partial \eta}
+ \cfrac{\partial(\hat\zeta_{x}\mathbf{\hat{f}})}{\partial \zeta}
+ \cfrac{\partial(\hat\tau_{x}\mathbf{\hat{f}}\equiv 0)}{\partial \tau}\\
&- \mathbf{\hat{f}}(\cfrac{\partial(\hat\xi_{x})}{\partial \xi}
+\cfrac{\partial(\hat\eta_{x})}{\partial \eta}
+\cfrac{\partial(\hat\zeta_{x})}{\partial \zeta}
)\\
&+ \cfrac{\partial(\hat\xi_{y} \mathbf{\hat{g}})}{\partial \xi}
+ \cfrac{\partial(\hat\eta_{y}\mathbf{\hat{g}})}{\partial \eta}
+ \cfrac{\partial(\hat\zeta_{y}\mathbf{\hat{g}})}{\partial \zeta}
+ \cfrac{\partial(\hat\tau_{y}\mathbf{\hat{g}}\equiv 0)}{\partial \tau}\\
&-\mathbf{\hat{g}}( \cfrac{\partial(\hat\xi_{y})}{\partial \xi}
+ \cfrac{\partial(\hat\eta_{y})}{\partial \eta}
+ \cfrac{\partial(\hat\zeta_{y})}{\partial \zeta})\\
&+ \cfrac{\partial(\hat\xi_{z}\mathbf{\hat{h}})}{\partial \xi}
+ \cfrac{\partial(\hat\eta_{z}\mathbf{\hat{h}})}{\partial \eta}
+ \cfrac{\partial(\hat\zeta_{z}\mathbf{\hat{h}})}{\partial \zeta}
+ \cfrac{\partial(\hat\tau_{z}\mathbf{\hat{h}}\equiv 0)}{\partial \tau}\\
&-\mathbf{\hat{h}}(\cfrac{\partial(\hat\xi_{z})}{\partial \xi}
+ \cfrac{\partial(\hat\eta_{z})}{\partial \eta}
+ \cfrac{\partial(\hat\zeta_{z})}{\partial \zeta})
\end{align}\end{split}\]
\[\begin{split}\begin{align}
&\cfrac{\partial(\hat\xi_{x})}{\partial \xi}
+\cfrac{\partial(\hat\eta_{x})}{\partial \eta}
+\cfrac{\partial(\hat\zeta_{x})}{\partial \zeta}\\
& = \cfrac{\partial({y}_{\eta}{z}_{\zeta}-{z}_{\eta}{y}_{\zeta})}{\partial \xi}
+\cfrac{\partial({z}_{\xi}{y}_{\zeta}-{y}_{\xi}{z}_{\zeta})}{\partial \eta}
+\cfrac{\partial({y}_{\xi}{z}_{\eta}-{z}_{\xi}{y}_{\eta})}{\partial \zeta}\\
&=({y}_{\xi\eta}{z}_{\zeta}+{y}_{\eta}{z}_{\xi\zeta}-{z}_{\xi\eta}{y}_{\zeta}-{z}_{\eta}{y}_{\xi\zeta})\\
&+({z}_{\xi\eta}{y}_{\zeta}+{z}_{\xi}{y}_{\eta\zeta}-{y}_{\xi\eta}{z}_{\zeta}-{y}_{\xi}{z}_{\eta\zeta})\\
&+({y}_{\xi\zeta}{z}_{\eta}+{y}_{\xi}{z}_{\eta\zeta}-{z}_{\xi\zeta}{y}_{\eta}-{z}_{\xi}{y}_{\eta\zeta})\\
&={y}_{\xi}({z}_{\eta\zeta}-{z}_{\eta\zeta})+{y}_{\eta}({z}_{\xi\zeta}-{z}_{\xi\zeta})+{y}_{\zeta}({z}_{\xi\eta}-{z}_{\xi\eta})\\
&+{z}_{\xi}({y}_{\eta\zeta}-{y}_{\eta\zeta})+{z}_{\eta}(-{y}_{\xi\zeta}+{y}_{\xi\zeta})+{z}_{\zeta}({y}_{\xi\eta}-{y}_{\xi\eta})\\
&=0
\end{align}\end{split}\]
\[\begin{split}\begin{align}
&\cfrac{\partial(\hat\xi_{y})}{\partial \xi}
+ \cfrac{\partial(\hat\eta_{y})}{\partial \eta}
+ \cfrac{\partial(\hat\zeta_{y})}{\partial \zeta}\\
&=\cfrac{\partial({z}_{\eta}{x}_{\zeta}-{x}_{\eta}{z}_{\zeta})}{\partial \xi}
+\cfrac{\partial({x}_{\xi}{z}_{\zeta}-{z}_{\xi}{x}_{\zeta})}{\partial \eta}
+\cfrac{\partial({z}_{\xi}{x}_{\eta}-{x}_{\xi}{z}_{\eta})}{\partial \zeta}\\
&={z}_{\xi\eta}{x}_{\zeta}+{z}_{\eta}{x}_{\xi\zeta}-{x}_{\xi\eta}{z}_{\zeta}-{x}_{\eta}{z}_{\xi\zeta}\\
&+{x}_{\xi\eta}{z}_{\zeta}+{x}_{\xi}{z}_{\eta\zeta}-{z}_{\xi\eta}{x}_{\zeta}-{z}_{\xi}{x}_{\eta\zeta}\\
&+{z}_{\xi\zeta}{x}_{\eta}+{z}_{\xi}{x}_{\eta\zeta}-{x}_{\xi\zeta}{z}_{\eta}-{x}_{\xi}{z}_{\eta\zeta}\\
&={x}_{\xi}({z}_{\eta\zeta}-{z}_{\eta\zeta})+{x}_{\eta}(-{z}_{\xi\zeta}+{z}_{\xi\zeta})+{x}_{\zeta}({z}_{\xi\eta}-{z}_{\xi\eta})\\
&+{z}_{\xi}(-{x}_{\eta\zeta}+{x}_{\eta\zeta})+{z}_{\eta}({x}_{\xi\zeta}-{x}_{\xi\zeta})+{z}_{\zeta}(-{x}_{\xi\eta}+{x}_{\xi\eta})\\
&=0
\end{align}\end{split}\]
\[\begin{split}\begin{align}
&\cfrac{\partial(\hat\xi_{z})}{\partial \xi}
+ \cfrac{\partial(\hat\eta_{z})}{\partial \eta}
+ \cfrac{\partial(\hat\zeta_{z})}{\partial \zeta}\\
&=\cfrac{\partial({x}_{\eta}{y}_{\zeta}-{y}_{\eta}{x}_{\zeta})}{\partial \xi}
+ \cfrac{\partial({y}_{\xi}{x}_{\zeta}-{x}_{\xi}{y}_{\zeta})}{\partial \eta}
+ \cfrac{\partial({x}_{\xi}{y}_{\eta}-{y}_{\xi}{x}_{\eta})}{\partial \zeta}\\
&={x}_{\xi\eta}{y}_{\zeta}+{x}_{\eta}{y}_{\xi\zeta}-{y}_{\xi\eta}{x}_{\zeta}-{y}_{\eta}{x}_{\xi\zeta}\\
&+{y}_{\xi\eta}{x}_{\zeta}+{y}_{\xi}{x}_{\eta\zeta}-{x}_{\xi\eta}{y}_{\zeta}-{x}_{\xi}{y}_{\eta\zeta}\\
&+{x}_{\xi\zeta}{y}_{\eta}+{x}_{\xi}{y}_{\eta\zeta}-{y}_{\xi\zeta}{x}_{\eta}-{y}_{\xi}{x}_{\eta\zeta}\\
&={x}_{\xi}(-{y}_{\eta\zeta}+{y}_{\eta\zeta})+{x}_{\eta}({y}_{\xi\zeta}-{y}_{\xi\zeta})+{x}_{\zeta}(-{y}_{\xi\eta}+{y}_{\xi\eta})\\
&+{y}_{\xi}({x}_{\eta\zeta}-{x}_{\eta\zeta})+{y}_{\eta}(-{x}_{\xi\zeta}+{x}_{\xi\zeta})+{y}_{\zeta}({x}_{\xi\eta}-{x}_{\xi\eta})\\
&=0
\end{align}\end{split}\]
\[\begin{split}\begin{align}
&\cfrac{\partial (\hat\xi_{t})}{\partial \xi}
+\cfrac{\partial(\hat\eta_{t})}{\partial \eta}
+\cfrac{\partial(\hat\zeta_{t})}{\partial \zeta}\\
&=\cfrac{\partial (-x_{\tau}\hat\xi_{x}-y_{\tau}\hat\xi_{y}-z_{\tau}\hat\xi_{z})}{\partial \xi}
+\cfrac{\partial(-x_{\tau}\hat\eta_{x}-y_{\tau}\hat\eta_{y}-z_{\tau}\hat\eta_{z})}{\partial \eta}
+\cfrac{\partial(-x_{\tau}\hat\zeta_{x}-y_{\tau}\hat\zeta_{y}-z_{\tau}\hat\zeta_{z})}{\partial \zeta}\\
&=-x_{\tau}(\cfrac{\partial\hat\xi_{x}}{\partial \xi}
+\cfrac{\partial\hat\eta_{x}}{\partial \eta}
+\cfrac{\partial\hat\zeta_{x}}{\partial \zeta})
-y_{\tau}(\cfrac{\partial\hat\xi_{y}}{\partial \xi}
+\cfrac{\partial\hat\eta_{y}}{\partial \eta}
+\cfrac{\partial\hat\zeta_{y}}{\partial \zeta})
-z_{\tau}(\cfrac{\partial\hat\xi_{z}}{\partial \xi}
+\cfrac{\partial\hat\eta_{z}}{\partial \eta}
+\cfrac{\partial\hat\zeta_{z}}{\partial \zeta})\\
&(-\hat\xi_{x}x_{\xi\tau}-\hat\xi_{y}y_{\xi\tau}-\hat\xi_{z}z_{\xi\tau})
+(-\hat\eta_{x}x_{\eta\tau}-\hat\eta_{y}y_{\eta\tau}-\hat\eta_{z}z_{\eta\tau})
+(-\hat\zeta_{x}x_{\zeta\tau}-\hat\zeta_{y}y_{\zeta\tau}-\hat\zeta_{z}z_{\zeta\tau})\\
&=(-\hat\xi_{x}x_{\xi\tau}-\hat\xi_{y}y_{\xi\tau}-\hat\xi_{z}z_{\xi\tau})
+(-\hat\eta_{x}x_{\eta\tau}-\hat\eta_{y}y_{\eta\tau}-\hat\eta_{z}z_{\eta\tau})
+(-\hat\zeta_{x}x_{\zeta\tau}-\hat\zeta_{y}y_{\zeta\tau}-\hat\zeta_{z}z_{\zeta\tau})
\end{align}\end{split}\]
\[\begin{split}\begin{align}
J^{-1}
& = {x}_{\xi}({y}_{\eta}{z}_{\zeta}-{z}_{\eta}{y}_{\zeta})
- {y}_{\xi}({x}_{\eta}{z}_{\zeta}-{z}_{\eta}{x}_{\zeta})
+ {z}_{\xi}({x}_{\eta}{y}_{\zeta}-{y}_{\eta}{x}_{\zeta}) \\
& = {x}_{\eta}({y}_{\zeta}{z}_{\xi}-{y}_{\xi}{z}_{\zeta})
+ {y}_{\eta}({x}_{\xi}{z}_{\zeta}-{z}_{\xi}{x}_{\zeta})
+ {z}_{\eta}({x}_{\zeta}{y}_{\xi}-{x}_{\xi}{y}_{\zeta}) \\
& = {x}_{\zeta}({y}_{\xi}{z}_{\eta}-{z}_{\xi}{y}_{\zeta})
+ {y}_{\zeta}({z}_{\xi}{x}_{\eta}-{x}_{\xi}{z}_{\eta})
+ {z}_{\zeta}({x}_{\xi}{y}_{\eta}-{y}_{\xi}{x}_{\eta}) \\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
J^{-1}
& = {x}_{\xi}({y}_{\eta}{z}_{\zeta}-{z}_{\eta}{y}_{\zeta})
+ {y}_{\xi}({z}_{\eta}{x}_{\zeta}-{x}_{\eta}{z}_{\zeta})
+ {z}_{\xi}({x}_{\eta}{y}_{\zeta}-{y}_{\eta}{x}_{\zeta}) \\
& = {x}_{\eta}({y}_{\zeta}{z}_{\xi}-{z}_{\zeta}{y}_{\xi})
+ {y}_{\eta}({z}_{\zeta}{x}_{\xi}-{x}_{\zeta}{z}_{\xi})
+ {z}_{\eta}({x}_{\zeta}{y}_{\xi}-{y}_{\zeta}{x}_{\xi}) \\
& = {x}_{\zeta}({y}_{\xi}{z}_{\eta}-{z}_{\xi}{y}_{\zeta})
+ {y}_{\zeta}({z}_{\xi}{x}_{\eta}-{x}_{\xi}{z}_{\eta})
+ {z}_{\zeta}({x}_{\xi}{y}_{\eta}-{y}_{\xi}{x}_{\eta}) \\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
J^{-1} & = x_{\xi}\hat\xi_{x}+y_{\xi}\hat\xi_{y}+z_{\xi}\hat\xi_{z}\\
& = x_{\eta}\hat\eta_{x}+y_{\eta}\hat\eta_{y}+z_{\eta}\hat\eta_{z}\\
& = x_{\zeta}\hat\zeta_{x}+y_{\zeta}\hat\zeta_{y}+z_{\zeta}\hat\zeta_{z}\\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
1 & = x_{\xi}\xi_{x}+y_{\xi}\xi_{y}+z_{\xi}\xi_{z}\\
& = x_{\eta}\eta_{x}+y_{\eta}\eta_{y}+z_{\eta}\eta_{z}\\
& = x_{\zeta}\zeta_{x}+y_{\zeta}\zeta_{y}+z_{\zeta}\zeta_{z}\\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\xi_{t} &= -x_{\tau}\xi_{x}-y_{\tau}\xi_{y}-z_{\tau}\xi_{z}\\
\eta_{t} &= -x_{\tau}\eta_{x}-y_{\tau}\eta_{y}-z_{\tau}\eta_{z}\\
\zeta_{t} &= -x_{\tau}\zeta_{x}-y_{\tau}\zeta_{y}-z_{\tau}\zeta_{z}\\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\hat\xi_{t} &= -x_{\tau}\hat\xi_{x}-y_{\tau}\hat\xi_{y}-z_{\tau}\hat\xi_{z}\\
\hat\eta_{t} &= -x_{\tau}\hat\eta_{x}-y_{\tau}\hat\eta_{y}-z_{\tau}\hat\eta_{z}\\
\hat\zeta_{t} &= -x_{\tau}\hat\zeta_{x}-y_{\tau}\hat\zeta_{y}-z_{\tau}\hat\zeta_{z}\\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
&\cfrac{\partial (\hat\xi_{t})}{\partial \xi}
+\cfrac{\partial(\hat\eta_{t})}{\partial \eta}
+\cfrac{\partial(\hat\zeta_{t})}{\partial \zeta}\\
&=(-\hat\xi_{x}x_{\xi\tau}-\hat\xi_{y}y_{\xi\tau}-\hat\xi_{z}z_{\xi\tau})
+(-\hat\eta_{x}x_{\eta\tau}-\hat\eta_{y}y_{\eta\tau}-\hat\eta_{z}z_{\eta\tau})
+(-\hat\zeta_{x}x_{\zeta\tau}-\hat\zeta_{y}y_{\zeta\tau}-\hat\zeta_{z}z_{\zeta\tau})
\end{align}\end{split}\]
\[\begin{split}\begin{array}{c}
\hat{\xi}_{x}=({y}_{\eta}{z}_{\zeta}-{z}_{\eta}{y}_{\zeta})\\
\hat{\xi}_{y}=({z}_{\eta}{x}_{\zeta}-{x}_{\eta}{z}_{\zeta})\\
\hat{\xi}_{z}=({x}_{\eta}{y}_{\zeta}-{y}_{\eta}{x}_{\zeta})\\
\hat{\xi}_{x\tau}=({y}_{\eta\tau}{z}_{\zeta}+{y}_{\eta}{z}_{\zeta\tau}-{z}_{\eta\tau}{y}_{\zeta}-{z}_{\eta}{y}_{\zeta\tau})\\
\hat{\xi}_{y\tau}=({z}_{\eta\tau}{x}_{\zeta}+{z}_{\eta}{x}_{\zeta\tau}-{x}_{\eta\tau}{z}_{\zeta}-{x}_{\eta}{z}_{\zeta\tau})\\
\hat{\xi}_{z\tau}=({x}_{\eta\tau}{y}_{\zeta}+{x}_{\eta}{y}_{\zeta\tau}-{y}_{\eta\tau}{x}_{\zeta}-{y}_{\eta}{x}_{\zeta\tau})\\
x_{\xi}\hat{\xi}_{x\tau}=x_{\xi}({y}_{\eta\tau}{z}_{\zeta}+{y}_{\eta}{z}_{\zeta\tau}-{z}_{\eta\tau}{y}_{\zeta}-{z}_{\eta}{y}_{\zeta\tau})\\
y_{\xi}\hat{\xi}_{y\tau}=y_{\xi}({z}_{\eta\tau}{x}_{\zeta}+{z}_{\eta}{x}_{\zeta\tau}-{x}_{\eta\tau}{z}_{\zeta}-{x}_{\eta}{z}_{\zeta\tau})\\
z_{\xi}\hat{\xi}_{z\tau}=z_{\xi}({x}_{\eta\tau}{y}_{\zeta}+{x}_{\eta}{y}_{\zeta\tau}-{y}_{\eta\tau}{x}_{\zeta}-{y}_{\eta}{x}_{\zeta\tau})\\
\end{array}\end{split}\]
\[\begin{split}\begin{align}
x_{\xi}\hat{\xi}_{x\tau}+y_{\xi}\hat{\xi}_{y\tau}+z_{\xi}\hat{\xi}_{z\tau}
& = x_{\xi}({y}_{\eta\tau}{z}_{\zeta}+{y}_{\eta}{z}_{\zeta\tau}-{z}_{\eta\tau}{y}_{\zeta}-{z}_{\eta}{y}_{\zeta\tau})\\
& + y_{\xi}({z}_{\eta\tau}{x}_{\zeta}+{z}_{\eta}{x}_{\zeta\tau}-{x}_{\eta\tau}{z}_{\zeta}-{x}_{\eta}{z}_{\zeta\tau})\\
& + z_{\xi}({x}_{\eta\tau}{y}_{\zeta}+{x}_{\eta}{y}_{\zeta\tau}-{y}_{\eta\tau}{x}_{\zeta}-{y}_{\eta}{x}_{\zeta\tau})\\
&={x}_{\eta\tau}({y}_{\zeta}z_{\xi}-{z}_{\zeta}y_{\xi})
+{y}_{\eta\tau}(x_{\xi}{z}_{\zeta}-z_{\xi}{x}_{\zeta})
+{z}_{\eta\tau}(y_{\xi}{x}_{\zeta}-x_{\xi}{y}_{\zeta})\\
&+{x}_{\zeta\tau}(y_{\xi}{z}_{\eta}-z_{\xi}{y}_{\eta})
+{y}_{\zeta\tau}(z_{\xi}{x}_{\eta}-x_{\xi}{z}_{\eta})
+{z}_{\zeta\tau}(x_{\xi}{y}_{\eta}-y_{\xi}{x}_{\eta})
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\hat\eta_{x} & = +({z}_{\xi}{y}_{\zeta}-{y}_{\xi}{z}_{\zeta})\\
\hat\eta_{y} & = +({x}_{\xi}{z}_{\zeta}-{z}_{\xi}{x}_{\zeta})\\
\hat\eta_{z} & = +({y}_{\xi}{x}_{\zeta}-{x}_{\xi}{y}_{\zeta})\\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\hat\zeta_{x} & = +({y}_{\xi}{z}_{\eta}-{z}_{\xi}{y}_{\eta})\\
\hat\zeta_{y} & = +({z}_{\xi}{x}_{\eta}-{x}_{\xi}{z}_{\eta})\\
\hat\zeta_{z} & = +({x}_{\xi}{y}_{\eta}-{y}_{\xi}{x}_{\eta})\\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
x_{\xi}\hat{\xi}_{x\tau}+y_{\xi}\hat{\xi}_{y\tau}+z_{\xi}\hat{\xi}_{z\tau}
&={x}_{\eta\tau}({y}_{\zeta}z_{\xi}-{z}_{\zeta}y_{\xi})
+{y}_{\eta\tau}(x_{\xi}{z}_{\zeta}-z_{\xi}{x}_{\zeta})
+{z}_{\eta\tau}(y_{\xi}{x}_{\zeta}-x_{\xi}{y}_{\zeta})\\
&+{x}_{\zeta\tau}(y_{\xi}{z}_{\eta}-z_{\xi}{y}_{\eta})
+{y}_{\zeta\tau}(z_{\xi}{x}_{\eta}-x_{\xi}{z}_{\eta})
+{z}_{\zeta\tau}(x_{\xi}{y}_{\eta}-y_{\xi}{x}_{\eta})\\
&={x}_{\eta\tau}(\hat\eta_{x})
+{y}_{\eta\tau}(\hat\eta_{y})
+{z}_{\eta\tau}(\hat\eta_{z})\\
&+{x}_{\zeta\tau}(\hat\zeta_{x})
+{y}_{\zeta\tau}(\hat\zeta_{y})
+{z}_{\zeta\tau}(\hat\zeta_{z})\\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
\cfrac{\partial J^{-1}}{\partial \tau}=\cfrac{\partial (x_{\xi}\hat\xi_{x}+y_{\xi}\hat\xi_{y}+z_{\xi}\hat\xi_{z})}{\partial \tau}
=(x_{\xi\tau}\hat\xi_{x}+y_{\xi\tau}\hat\xi_{y}+z_{\xi\tau}\hat\xi_{z})
+(x_{\xi}\hat\xi_{x\tau}+y_{\xi}\hat\xi_{y\tau}+z_{\xi}\hat\xi_{z\tau})\\
\end{align}\end{split}\]
\[\begin{split}\begin{align}
&\cfrac{\partial J^{-1}}{\partial \tau}+ \cfrac{\partial (\hat\xi_{t})}{\partial \xi}
+\cfrac{\partial(\hat\eta_{t})}{\partial \eta}
+\cfrac{\partial(\hat\zeta_{t})}{\partial \zeta}\\
& = (x_{\xi\tau}\hat\xi_{x}+y_{\xi\tau}\hat\xi_{y}+z_{\xi\tau}\hat\xi_{z})
+(x_{\xi}\hat\xi_{x\tau}+y_{\xi}\hat\xi_{y\tau}+z_{\xi}\hat\xi_{z\tau})\\
& + (-\hat\xi_{x}x_{\xi\tau}-\hat\xi_{y}y_{\xi\tau}-\hat\xi_{z}z_{\xi\tau})
+(-\hat\eta_{x}x_{\eta\tau}-\hat\eta_{y}y_{\eta\tau}-\hat\eta_{z}z_{\eta\tau})
+(-\hat\zeta_{x}x_{\zeta\tau}-\hat\zeta_{y}y_{\zeta\tau}-\hat\zeta_{z}z_{\zeta\tau})\\
&=(x_{\xi}\hat\xi_{x\tau}+y_{\xi}\hat\xi_{y\tau}+z_{\xi}\hat\xi_{z\tau})+(-\hat\eta_{x}x_{\eta\tau}-\hat\eta_{y}y_{\eta\tau}-\hat\eta_{z}z_{\eta\tau})
+(-\hat\zeta_{x}x_{\zeta\tau}-\hat\zeta_{y}y_{\zeta\tau}-\hat\zeta_{z}z_{\zeta\tau})\\
&=0
\end{align}\end{split}\]
Therefor the general curvilinear equation can now be expressed:
\[\begin{split}\begin{align}
\cfrac{\partial(J^{-1}\hat{\mathbf{q}})}{\partial t}
&+\cfrac{\partial}{\partial \xi}[\hat\xi_{t}\hat{\mathbf{q}}+\hat\xi_{x}\hat{\mathbf{f}}+\hat\xi_{y}\hat{\mathbf{g}}+\hat\xi_{z}\hat{\mathbf{h}}]\\
&+\cfrac{\partial}{\partial \eta}[\hat\eta_{t}\hat{\mathbf{q}}+\hat\eta_{x}\hat{\mathbf{f}}+\hat\eta_{y}\hat{\mathbf{g}}+\hat\eta_{z}\hat{\mathbf{h}}]\\
&+\cfrac{\partial}{\partial \zeta}[\hat\zeta_{t}\hat{\mathbf{q}}+\hat\zeta_{x}\hat{\mathbf{f}}+\hat\zeta_{y}\hat{\mathbf{g}}+\hat\zeta_{z}\hat{\mathbf{h}}]\\
&=0
\end{align}\end{split}\]
or in the more compact form:
\[\begin{align}
\cfrac{\partial(J^{-1}{\mathbf{Q}})}{\partial \tau}
+\cfrac{\partial{\mathbf{F}}}{\partial \xi}
+\cfrac{\partial{\mathbf{G}}}{\partial \eta}
+\cfrac{\partial{\mathbf{H}}}{\partial \zeta}
=0
\end{align}\]
where the vector of conserved variables, \(\mathbf{Q}\), and the vector of inviscid flux terms, \(\mathbf{F}\), \(\mathbf{G}\) and \(\mathbf{H}\), are:
\[\begin{split}\mathbf{Q}=\mathbf{\hat{q}}=\begin{bmatrix}
\rho\\ \rho u \\ \rho v \\ \rho w\\ \rho E
\end{bmatrix}\end{split}\]
\[\begin{split}\mathbf{F}=\hat\xi_{t}\hat{\mathbf{q}}+\hat\xi_{x}\hat{\mathbf{f}}+\hat\xi_{y}\hat{\mathbf{g}}+\hat\xi_{z}\hat{\mathbf{h}}=\begin{bmatrix}
\rho U\\
\rho u U + \hat\xi_{x}p\\
\rho v U + \hat\xi_{y}p\\
\rho w U + \hat\xi_{z}p\\
\rho H U - \hat\xi_{t}p\\
\end{bmatrix}\end{split}\]
\[\begin{split}\mathbf{G}=\hat\eta_{t}\hat{\mathbf{q}}+\hat\eta_{x}\hat{\mathbf{f}}+\hat\eta_{y}\hat{\mathbf{g}}+\hat\eta_{z}\hat{\mathbf{h}}=\begin{bmatrix}
\rho V\\
\rho u V + \hat\eta_{x}p\\
\rho v V + \hat\eta_{y}p\\
\rho w V + \hat\eta_{z}p\\
\rho H V - \hat\eta_{t}p\\
\end{bmatrix}\end{split}\]
\[\begin{split}\mathbf{H}=\hat\zeta_{t}\hat{\mathbf{q}}+\hat\zeta_{x}\hat{\mathbf{f}}+\hat\zeta_{y}\hat{\mathbf{g}}+\hat\zeta_{z}\hat{\mathbf{h}}=\begin{bmatrix}
\rho W\\
\rho u W + \hat\zeta_{x}p\\
\rho v W + \hat\zeta_{y}p\\
\rho w W + \hat\zeta_{z}p\\
\rho H W - \hat\zeta_{t}p\\
\end{bmatrix}\end{split}\]
\(U\), \(V\) and \(W\) are the contravariant velocity components in the \(\xi\), \(\eta\) and \(\zeta\) directions and are defined as:
\[\begin{split}\begin{align}
U & = \hat\xi_{x}u+\hat\xi_{y}v+\hat\xi_{z}w+\hat\xi_{t}\\
V & = \hat\eta_{x}u+\hat\eta_{y}v+\hat\eta_{z}w+\hat\eta_{t}\\
W & = \hat\zeta_{x}u+\hat\zeta_{y}v+\hat\zeta_{z}w+\hat\zeta_{t}\\
\end{align}\end{split}\]