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:

Let $C$ be a directed smooth curve in the complex plane $\C$.

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

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

$\rho: \closedint a b \to \C: \map \rho t = \map \gamma {a + b - t}$

From Reversed Directed Smooth Curve is Directed Smooth Curve, it follows that $-C$ is a directed smooth curve.

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 $\R^n$.

Sources

• 2001: Christian Berg: Kompleks funktionsteori: $\S 2.2$