# Definition:Curvilinear Coordinate System

## 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.