Length of Element of Arc in Orthogonal Curvilinear Coordinates

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:

where $\tuple {x, y, z}$ denotes the Cartesian coordinates.

Let $S$ be an infinitesimal arc.

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

Then:

where:


 * $\d q_i$ is the projection of $S$ onto the unit normal to the curvilinear coordinate surface determined by $q_i$, for $i \in \set {1, 3}$


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