Definition:Curvilinear Coordinate System/Complex Plane

Definition

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 $f$.

Coordinate Curves

Let $c_1$ and $c_2$ be constants.

The curves:

$\map u {x, y} = c_1$

$\map v {x, y} = c_2$

are the coordinate curves of $f$.

Historical Note

Curvilinear coordinates were first introduced by Carl Friedrich Gauss in his $1827$ work Disquisitiones Generales circa Superficies Curvas.

This was as a result of his commission to perform a geodetic survey of the Kingdom of Hanover from around $1820$ onwards.

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: Curvilinear Coordinates
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.25$: Gauss ($\text {1777}$ – $\text {1855}$)