Length of Element of Arc in Orthogonal Curvilinear Coordinates

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Theorem

Let $\tuple {q_1, q_2, q_3}$ denote a set of orthogonal curvilinear coordinates.

Let the relation between those orthogonal 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.


Let $S$ be an infinitesimal arc.

Let $\d s$ be the length of $S$


Then:

\(\ds \d s^2\) \(=\) \(\ds {h_1}^2 {\rd q_1}^2 + {h_2}^2 {\rd q_2}^2 + {h_3}^2 {\rd q_3}^2\)
\(\ds \) \(=\) \(\ds \sum_i {h_i}^2 {\rd q_i}^2\) for $i \in \set {1, 2, 3}$


where:

$\d q_i$ is the projection of $S$ onto the unit normal to the curvilinear coordinate surface determined by $q_i$, for $i \in \set {1, 3}$
${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$


Proof

By definition of the metric of $\tuple {q_1, q_2, q_3}$:

\(\ds \d s^2\) \(=\) \(\ds \d x^2 + \d y^2 + \d z^2\)
\(\ds \) \(=\) \(\ds \sum_{i, j} {h_{i j} }^2 \rd q_i \rd q_j\) for $i, j \in \set {1, 2, 3}$

From Value of Curvilinear Coordinate Metric:

$\forall i, j \in \set {1, 2, 3}: {h_{i j} }^2 = \dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j}$


But we have that $\tuple {q_1, q_2, q_3}$ is orthogonal.

From Definition 1 of orthogonal curvilinear coordinates:

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

wherever $i \ne j$.

Hence we have:

\(\ds \d s^2\) \(=\) \(\ds {h_{1 1} }^2 {\rd q_1}^2 + {h_{2 2} }^2 {\rd q_2}^2 + {h_{3 3} }^2 {\rd q_3}^2\)
\(\ds \) \(=\) \(\ds \sum_i {h_{i i} }^2 {\rd q_i}^2\) for $i \in \set {1, 2, 3}$

Elements $h_{i i}$ are those elements of the metric which do not vanish when $\tuple {q_1, q_2, q_3}$ is orthogonal.

To streamline notation, we rename them $h_1$, $h_2$ and $h_3$.


Hence:

\(\ds \d s^2\) \(=\) \(\ds {h_1}^2 {\rd q_1}^2 + {h_2}^2 {\rd q_2}^2 + {h_3}^2 {\rd q_3}^2\)
\(\ds \) \(=\) \(\ds \sum_i {h_i}^2 {\rd q_i}^2\) for $i \in \set {1, 2, 3}$

where:

${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$

$\blacksquare$


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