Category:Orientation of Complex Contour
This category contains results about Orientation of Complex Contour.
Definitions specific to this category can be found in Definitions/Orientation of Complex Contour.
Let $C$ be a 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 $t$ } }$.
Positively Oriented Contour
Let $D \subseteq \C$ be a connected domain.
Let $\Img C \subseteq \partial D$, where $\Img C$ denotes the image of $C$, and $\partial D$ denotes the boundary of $D$.
Then $C$ is positively oriented with respect to $D$, if and only if for all $t \in \openint a b \setminus K$, there exists $r \in \R_{>0}$ such that:
- for all $\epsilon \in \openint 0 r$ : $\map \gamma t + \epsilon i \map {\gamma'} t \in D$, and $\map \gamma t - \epsilon i \map {\gamma'} t \notin D$
Positively Oriented Simple Closed Contour
Let $C$ be a simple closed contour in the complex plane $\C$ with parameterization $\gamma: \closedint a b \to \C$.
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 $\epsilon \in \openint 0 r$ : $\map \gamma t + \epsilon i \map {\gamma'} t \in \Int C$
Negatively Oriented Contour
Let $D \subseteq \C$ be a connected domain.
Let $\Img C \subseteq \partial D$, where $\Img C$ denotes the image of $C$, and $\partial D$ denotes the boundary of $D$.
Then $C$ is negatively oriented with respect to $D$, if and only if for all $t \in \openint a b \setminus K$, there exists $r \in \R_{>0}$ such that:
- for all $\epsilon \in \openint 0 r$ : $\map \gamma t - \epsilon i \map {\gamma'} t \in D$, and $\map \gamma t + \epsilon i \map {\gamma'} t \notin D$
Negatively Oriented Simple Closed Contour
Let $C$ be a simple closed contour in the complex plane $\C$ with parameterization $\gamma: \closedint a b \to \C$.
Let $\Int C$ denote the interior of $C$
Then $C$ is negatively oriented, if and only if for all $t \in \openint a b \setminus K$, there exists $r \in \R_{>0}$ such that:
- for all $\epsilon \in \openint 0 r$ : $\map \gamma t - \epsilon i \map {\gamma'} t \in \Int C$
Subcategories
This category has only the following subcategory.
Pages in category "Orientation of Complex Contour"
The following 9 pages are in this category, out of 9 total.