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.


The length $\d l$ of a small arc is given by:

$\d l = {h_1}^2 {\d q_1}^2 + {h_2}^2 {\d q_2}^2 + {h_3}^2 {\d q_3}^2$

where:

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


Proof


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