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

• Definition:Rectilinear Coordinate System

• Results about curvilinear coordinate systems can be found here.

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Coordinates: $2$. Coordinates
• 1961: Ian N. Sneddon: Special Functions of Mathematical Physics and Chemistry (2nd ed.) ... (previous) ... (next): Chapter $\text I$: Introduction: $\S 1$. The origin of special functions