\begin{definition}[Definition:Complex Curvilinear Coordinates/Coordinate Curves]
Let $u + i v = \map f {x + i y}$ be a complex transformation.
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 $f$.
\end{definition}