Definition:Reversed Directed Smooth Curve

Definition
Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ be a directed smooth curves in $\R^n$.

Let $C$ be parameterized by the smooth path $\gamma: \closedint a b \to \R^n$.

The reversed directed smooth curve of $C$ is denoted $-C$ and is defined as the directed smooth curve that is parameterized by:


 * $\rho: \closedint a b \to \R^n: \map \rho t = \map \gamma {a + b - t}$

Complex Plane
The definition carries over to the complex plane, in which context it is usually applied:

Also known as
A reversed directed smooth curve is called a reversed curve or a reciprocal curve in some texts.

Also denoted as
The reversed directed smooth curve of $C$ is denotes as $C^-$ in some texts.

Also see

 * Reversed Directed Smooth Curve is Directed Smooth Curve: a demonstration that $-C$ is a directed smooth curve in $\R^n$.