# 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.