# Definition:Curvilinear Coordinates

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## 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 ($\text {1777}$ – $\text {1855}$)