Definition:Orthogonal Coordinate System

Definition

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

Orthogonal Curvilinear Coordinates

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.

Definition 1

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.

Definition 2

Let $\KK$ have the property that for every arbitrary pair of coordinate surfaces $q_i \in \QQ_i$ and $q_j \in \QQ_j$ where $i \ne j$:

$q_i$ and $q_j$ are orthogonal.

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

Rectangular Coordinate System

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