Definition:Orthogonal Coordinate System

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Definition

An orthogonal coordinate system is a coordinate system in which the coordinate axes are pairwise perpendicular.


Orthogonal Curvilinear Coordinates

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


Let these equations have the property that:

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

wherever $i \ne j$.

Then $\tuple {q_1, q_2, q_3}$ are orthogonal curvilinear coordinates.


Rectangular Coordinate System

A rectangular coordinate system is a Cartesian coordinate system in which each of the coordinate axes are perpendicular to each other.