Path as Parameterization of Contour

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Theorem

Let $\closedint a b$ be a closed real interval.

Let $\gamma: \closedint a b \to \C$ be a path.

Let there exist $n \in \N$ and a subdivision $\set {a_0, a_1, \ldots, a_n}$ of $\closedint a b$ such that:

$\gamma {\restriction_{\closedint {a_{k - 1} } {a_k} } }$ is a smooth path for all $k \in \set {1, \ldots, n}$

where $\gamma {\restriction_{\closedint {a_{k - 1} } {a_k} } }$ denotes the restriction of $\gamma$ to $\closedint {a_{k - 1} } {a_k}$.

Then there exists a contour $C$ with parameterization $\gamma$.

Corollary 1

If $\gamma$ is a closed path, then $C$ is a closed contour.

Corollary 2

If $\gamma$ is a Jordan arc, then $C$ is a simple contour, and if $\gamma$ is a Jordan curve, then $C$ is a simple closed contour.

Proof

Put $\gamma_k = \gamma {\restriction_{\closedint {a_{k - 1} } {a_k} } } : \closedint {a_{k - 1} } {a_k} \to \C$.

By definition, it follows that there exists a directed smooth curve $C_k$ with parameterization $\gamma_k$.

For all $k \in \set {1, \ldots, n - 1}$, we have:

$\map {\gamma_k} {a_k} = \map {\gamma_{k + 1} } {a_k}$

Define the contour $C$ as the concatenation $C_1 \cup C_2 \cup \ldots \cup C_n$.

Then, it follows by definition that $\gamma$ is a possible parameterization of $C$.

$\blacksquare$