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 $\tuple {q_1, q_2, q_3}$ 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.