Category:Definitions/Directed Smooth Curves (Complex Plane)

This category contains definitions related to Directed Smooth Curves (Complex Plane).
Related results can be found in Category:Directed Smooth Curves (Complex Plane).

Let $\gamma : \closedint a b \to \C$ be a smooth path in $\C$.

The directed smooth curve with parameterization $\gamma$ is defined as an equivalence class of smooth paths as follows:

A smooth path $\sigma: \closedint c d \to \C$ belongs to the equivalence class of $\gamma$ if and only if:

there exists a bijective differentiable strictly increasing real function:

$\phi: \closedint c d \to \closedint a b$

such that $\sigma = \gamma \circ \phi$.