Rectilinear coordinates

Note

Normalised representations will not be discussed since they coincide with the Cartesian one.

The rectilinear coordinates are defined as

\[ \begin{align}\begin{aligned}& X^1 \equiv f_1 \left( x^1 \right),\\& X^2 \equiv f_2 \left( x^2 \right),\\& X^3 \equiv f_3 \left( x^3 \right).\end{aligned}\end{align} \]

\(f_i\) are not explicitly discussed here, which are to be determined when the computational grid is designed. Without loss of generality, I assume that the mapping functions \(f_i\) monotonically increase.

The transformation matrix is

\[\begin{split}\begin{pmatrix} \pder{x^1}{X^1} & \pder{x^1}{X^2} & \pder{x^1}{X^3} \\ \pder{x^2}{X^1} & \pder{x^2}{X^2} & \pder{x^2}{X^3} \\ \pder{x^3}{X^1} & \pder{x^3}{X^2} & \pder{x^3}{X^3} \\ \end{pmatrix} = \begin{pmatrix} \pder{x^1}{X^1} & 0 & 0 \\ 0 & \pder{x^2}{X^2} & 0 \\ 0 & 0 & \pder{x^3}{X^3} \\ \end{pmatrix} = \begin{pmatrix} H_1 & 0 & 0 \\ 0 & H_2 & 0 \\ 0 & 0 & H_3 \\ \end{pmatrix},\end{split}\]

or

\[\begin{split}\begin{pmatrix} \pder{X^1}{x^1} & \pder{X^1}{x^2} & \pder{X^1}{x^3} \\ \pder{X^2}{x^1} & \pder{X^2}{x^2} & \pder{X^2}{x^3} \\ \pder{X^3}{x^1} & \pder{X^3}{x^2} & \pder{X^3}{x^3} \\ \end{pmatrix} = \begin{pmatrix} \pder{X^1}{x^1} & 0 & 0 \\ 0 & \pder{X^2}{x^2} & 0 \\ 0 & 0 & \pder{X^3}{x^3} \\ \end{pmatrix} = \begin{pmatrix} \frac{1}{H_1} & 0 & 0 \\ 0 & \frac{1}{H_2} & 0 \\ 0 & 0 & \frac{1}{H_3} \\ \end{pmatrix}.\end{split}\]

As a consequence, the basis vectors are related by

\[ \begin{align}\begin{aligned}& \vec{E}_1 = \pder{x^1}{X^1} \vec{e}_1 = H_1 \vec{e}_1,\\& \vec{E}_2 = \pder{x^2}{X^2} \vec{e}_2 = H_2 \vec{e}_2,\\& \vec{E}_3 = \pder{x^3}{X^3} \vec{e}_3 = H_3 \vec{e}_3,\end{aligned}\end{align} \]

or

\[ \begin{align}\begin{aligned}& \vec{e}_1 = \pder{X^1}{x^1} \vec{E}_1 = \frac{1}{H_1} \vec{E}_1,\\& \vec{e}_2 = \pder{X^2}{x^2} \vec{E}_2 = \frac{1}{H_2} \vec{E}_2,\\& \vec{e}_3 = \pder{X^3}{x^3} \vec{E}_3 = \frac{1}{H_3} \vec{E}_3.\end{aligned}\end{align} \]

Although already introduced, the scale factors are

\[H_i = \sqrt{ \vec{E}_i \cdot \vec{E}_i } = \pder{x^i}{X^i},\]

and their spatial derivatives are

\[\begin{split}\begin{pmatrix} \pder{H_1}{X^1} & \pder{H_2}{X^1} & \pder{H_3}{X^1} \\ \pder{H_1}{X^2} & \pder{H_2}{X^2} & \pder{H_3}{X^2} \\ \pder{H_1}{X^3} & \pder{H_2}{X^3} & \pder{H_3}{X^3} \\ \end{pmatrix} = \begin{pmatrix} \pder{H_1}{X^1} & 0 & 0 \\ 0 & \pder{H_2}{X^2} & 0 \\ 0 & 0 & \pder{H_3}{X^3} \\ \end{pmatrix}.\end{split}\]

The Jacobian determinant is

\[J = H_1 H_2 H_3 = \pder{x^1}{X^1} \pder{x^2}{X^2} \pder{x^3}{X^3}.\]

Although not limited, I consider the velocity vector

\[\vec{u} \equiv \tder{\vec{x}}{t} = \sum_i u^i \vec{e}_i\]

as an example to see how the components are related. Then the contravariant components are

\[ \begin{align}\begin{aligned}& U^1 = \pder{X^1}{x^1} u^1 = \frac{u^1}{H_1},\\& U^2 = \pder{X^2}{x^2} u^2 = \frac{u^2}{H_2},\\& U^3 = \pder{X^3}{x^3} u^3 = \frac{u^3}{H_3}.\end{aligned}\end{align} \]