Definition:Curvilinear Coordinate System

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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:

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

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


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


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.