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Distributed Climate Science and Computing

The Cross Product in Non Orthogonal Coordinate Systems

The form of the cross product I’ve shown in my post Coriolis Forces is:

\boldsymbol{\Omega \times v} =  \begin{pmatrix} \Omega_y v_z - \Omega_z v_y \\ \Omega_z v_x - \Omega_x v_z  \\ \Omega_x v_y - \Omega_y v_x  \end{pmatrix}

The components of this cross product can be written as follows:

\left[ \boldsymbol{\Omega \times v} \right]_x=  \left[ \begin{matrix}\Omega_x&\Omega_y&\Omega_z \end{matrix} \right]\left[ \begin{matrix}0&0&0 \\ 0&0&1 \\ 0&-1&0\end{matrix} \right]\left[ \begin{matrix}v_x\\v_y\\v_z \end{matrix} \right]

\left[ \boldsymbol{\Omega \times v} \right]_y=  \left[ \begin{matrix}\Omega_x&\Omega_y&\Omega_z \end{matrix} \right]\left[ \begin{matrix}0&0&-1 \\ 0&0&0 \\ 1&0&0\end{matrix} \right]\left[ \begin{matrix}v_x\\v_y\\v_z \end{matrix} \right]

\left[ \boldsymbol{\Omega \times v} \right]_z=  \left[ \begin{matrix}\Omega_x&\Omega_y&\Omega_z \end{matrix} \right]\left[ \begin{matrix}0&1&0 \\ -1&0&0 \\ 0&0&0\end{matrix} \right]\left[ \begin{matrix}v_x\\v_y\\v_z \end{matrix} \right]

We will abbreviate these relationships as follows:

\left[ \boldsymbol{\Omega \times v} \right]_x=\Omega^T[\times]_xv

\left[ \boldsymbol{\Omega \times v} \right]_y=\Omega^T[\times]_yv

\left[ \boldsymbol{\Omega \times v} \right]_z=\Omega^T[\times]_zv

Now define the coordinate transform:

\boldsymbol{r'}=T \boldsymbol{r}
where
\boldsymbol{r}=\left[\begin{matrix}  x \\ y \\ z \end{matrix} \right]

Then the cross product components can be written as follows:

\left[ \boldsymbol{\Omega \times v} \right]_x=(\Omega^TT^T)((T^{-1})^T[\times]_xT^{-1})(Tv)

\left[ \boldsymbol{\Omega \times v} \right]_y=(\Omega^TT^T)((T^{-1})^T[\times]_yT^{-1})(Tv)

\left[ \boldsymbol{\Omega \times v} \right]_z=(\Omega^TT^T)((T^{-1})^T[\times]_zT^{-1})(Tv)

Now Right multiplying the matrix \left[ \boldsymbol{\Omega \times v} \right] by the transform T gives:

\left[ \boldsymbol{\Omega \times v} \right]_{x'}=\sum_{k \in \{x,y,z\}}T_{1,k}(\Omega^TT^T)((T^{-1})^T[\times]_kT^{-1})(Tv)

\left[ \boldsymbol{\Omega \times v} \right]_{y'}=\sum_{k \in \{x,y,z\}}T_{2,k}(\Omega^TT^T)((T^{-1})^T[\times]_kT^{-1})(Tv)

\left[ \boldsymbol{\Omega \times v} \right]_{z'}=\sum_{k \in \{x,y,z\}}T_{3,k}(\Omega^TT^T)((T^{-1})^T[\times]_kT^{-1})(Tv)

Which can be written in this form:
\left[ \boldsymbol{\Omega \times v} \right]_{x'}=\Omega'^T[\times]_{x'}v'

\left[ \boldsymbol{\Omega \times v} \right]_{y'}=\Omega'^T[\times]_{y'}{v'}

\left[ \boldsymbol{\Omega \times v} \right]_{z'}=\Omega^T[\times]_{z'}{v'}

Where:

\left[[\times]_{x'}\right]_{i,j}=\sum_{k \in \{x,y,z\}}T_{1,k}[((T^{-1})^T[\times]_kT^{-1})]_{i,j}

\left[[\times]_{y'}\right]_{i,j}=\sum_{k \in \{x,y,z\}}T_{2,k}[((T^{-1})^T[\times]_kT^{-1})]_{i,j}

\left[[\times]_{z'}\right]_{i,j}=\sum_{k \in \{x,y,z\}}T_{3,k}[((T^{-1})^T[\times]_kT^{-1})]_{i,j}

\Omega'=T\Omega
v'=Tv

September 7, 2009 Posted by | Math | 2 Comments