User:Anghel/Sandbox

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

Let $_0t \in \openint a b$ such that $\gamma$ is complex-differentiable at $t_0$.

Let $r \in \R_{>0}$ such that for all $\epsilon \in \openint 0 r$, we have:


 * $\map \gamma {t_0} + \epsilon i S \map {\gamma '}{ t_0 } \in \Int C$

with $S \in \set {1, -1}$, and $\Int C$ denotes the interior of $C$.

If $S = 1$, then $C$ is positively oriented, and if $S = -1$, then $C$ is negatively oriented.

Proof
Complex Plane is Homeomorphic to Real Plane shows that we can identify $\C$ with the Euclidean plane $\R^2$.

By Interior of Simple Closed Contour is Well-Defined, we can identify the image $\Img C$ with the image of a Jordan curve in $\R^2$.

Let $t \in \openint a b$ such that $\gamma$ is differentiable at $t$.

From Normal Vectors Form Space around Complex Contour, it follows that

Category:Complex Contour Integrals