Definition:Orthogonal Curvilinear Coordinates/Definition 1

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Definition

Let $\KK$ be a curvilinear coordinate system in $3$-space.

Let $\QQ_1$, $\QQ_2$ and $\QQ_3$ denote the one-parameter families that define the curvilinear coordinates.


Let the relation between those curvilinear coordinates and Cartesian coordinates be expressed as:

\(\ds x\) \(=\) \(\ds \map x {q_1, q_2, q_3}\)
\(\ds y\) \(=\) \(\ds \map y {q_1, q_2, q_3}\)
\(\ds z\) \(=\) \(\ds \map z {q_1, q_2, q_3}\)

where:

$\tuple {x, y, z}$ denotes the Cartesian coordinates of an arbitrary point $P$
$\tuple {q_1, q_2, q_3}$ denotes the curvilinear coordinates of $P$.


Let these equations have the property that the metric of $\KK$ between coordinate surfaces of $\QQ_i$ and $\QQ_j$ is zero where $i \ne j$.


That is, for every point $P$ expressible as $\tuple {x, y, z}$ and $\tuple {q_1, q_2, q_3}$:

$\dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j} = 0$

wherever $i \ne j$.


Then $\KK$ is an orthogonal curvilinear coordinate system.


Also see

  • Results about orthogonal curvilinear coordinates can be found here.


Sources

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