\begin{definition}[Definition:Curvilinear Coordinate System/Complex Plane]
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 $f$'''.
\end{definition}