Definition:Curvilinear Coordinate System
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
- 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