Definition:Curvilinear Coordinate System

From ProofWiki
Jump to navigation Jump to search

Definition

A coordinate system such that at least one of the coordinate axes is a curved line is called a system of curvilinear coordinates.


Cartesian Representation

The relation between curvilinear coordinates and Cartesian coordinates can be expressed as:

\(\ds x\) \(=\) \(\ds \map x {q_1, q_2, q_3}\)
\(\ds y\) \(=\) \(\ds \map y {q_1, q_2, q_3}\)
\(\ds z\) \(=\) \(\ds \map z {q_1, q_2, q_3}\)


\(\ds q_1\) \(=\) \(\ds \map {q_1} {x, y, z}\)
\(\ds q_2\) \(=\) \(\ds \map {q_2} {x, y, z}\)
\(\ds q_3\) \(=\) \(\ds \map {q_3} {x, y, z}\)


where:

$\tuple {x, y, z}$ denotes the Cartesian coordinates
$\tuple {q_1, q_2, q_3}$ denotes their curvilinear equivalents.


Examples

Polar Coordinates

The canonical example of a curvilinear coordinate system is the polar coordinate system.


Complex Curvilinear Coordinates

Let $u + i v = \map f {x + i y}$ be a complex transformation.

Let $P = \tuple {x, y}$ be a point in the complex plane.


Then $\tuple {\map u {x, y}, \map v {x, y} }$ are the curvilinear coordinates of $P$ under $f$.


Also see

  • Results about curvilinear coordinate systems can be found here.


Sources