Path as Parameterization of Contour

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Theorem

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a path.

Let there exist $n \in \N$ and a subdivision $\left\{{a_0, a_1, \ldots, a_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$ such that:

$\gamma {\restriction_{ \left[{a_{k - 1} \,.\,.\, a_k}\right] } }$ is a smooth path for all $k \in \left\{ {1, \ldots, n}\right\}$

where $\gamma {\restriction_{\left[{a_{k - 1} \,.\,.\, a_k}\right]} }$ denotes the restriction of $\gamma$ to $\left[{a_{k - 1} \,.\,.\, a_k}\right]$.


Then there exists a contour $C$ with parameterization $\gamma$ and these properties:

$(1): \quad$ If $\gamma$ is a closed path, then $C$ is a closed contour.
$(2): \quad$ If $\gamma$ is a Jordan arc, then $C$ is a simple contour.
$(3): \quad$ If $\gamma$ is a Jordan curve, then $C$ is a simple closed contour.


Proof

Put $\gamma_k = \gamma {\restriction_{ \left[{a_{k-1} \,.\,.\, a_k}\right] } } : \left[{a_{k-1} \,.\,.\, a_k}\right] \to \C$.

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

For all $k \in \left\{ {1, \ldots, n-1}\right\}$, we have $\gamma_k \left({a_k}\right) = \gamma_{k + 1} \left({a_k}\right)$.

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 $C$.

$\Box$


Suppose that $\gamma$ is a closed path.

Then $\gamma \left({a}\right) = \gamma_1 \left({a_0}\right) = \gamma_n \left({a_n}\right)$, so $C_1$ has start point $\gamma \left({a}\right)$, and $C_n$ has end point $\gamma \left({a}\right)$.

By definition, it follows that $C$ is a closed contour.

$\Box$


Suppose that $\gamma$ is a Jordan arc.

Let $k_1, k_2 \in \left\{ {1, \ldots, n}\right\}$, and $t_1 \in \left[{a_{k_1 - 1} \,.\,.\, a_{k_1} }\right), t_2 \in \left[{a_{k_2 - 1} \,.\,.\, a_{k_2} }\right)$.

Then $\gamma \left({t_1}\right) \ne \gamma \left({t_2}\right)$ by the definition of Jordan arc, so $\gamma_{k_1} \left({t_1}\right) \ne \gamma_{k_2} \left({t_2}\right)$.

Let instead $k \in \left\{ {1, \ldots, n}\right\}$ and $t \in \left[{a_{k-1} \,.\,.\, a_k}\right)$ with $t \ne a_1$.

Then $\gamma \left({t}\right) \ne \gamma \left({a_n}\right)$ by the definition of Jordan arc, so $\gamma_k \left({t}\right) \ne \gamma_n \left({a_n}\right)$.

By definition, it follows that $C$ is a simple contour.

$\Box$


Suppose that $\gamma$ is a Jordan curve.

As a Jordan curve by definition is both a Jordan arc and a closed path, it follows from what is shown above that $C$ is a simple closed contour.

$\blacksquare$