Definition:Orientation of Contour (Complex Plane)/Positive/Simple Closed

Definition

Let $C$ be a simple closed contour in the complex plane $\C$ with parameterization $\gamma: \closedint a b \to \C$.

Set $K := \set { t \in \closedint a b : \textrm{ $\gamma$ is not differentiable at $t$ } }$.

Let $\Int C$ denote the interior of $C$

Then $C$ is positively oriented, if and only if for all $t \in \openint a b \setminus K$, there exists $r \in \R_{>0}$ such that:

for all $\epsilon \in \openint 0 r$ : $\map \gamma t + \epsilon i \map {\gamma'} t \in \Int C$

Alternatively, we say that $C$ has a positive orientation.

Informally, we say that $\Int C$ lies to the left of the tangent vector $\map {\gamma'} t$.

Some texts say that $C$ is counterclockwise oriented, or anticlockwise oriented.

Some texts use the hyphenated form positively-oriented.