Reversed Contour is Contour

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Theorem

Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ be a contour in $\R^n$ that is defined as a concatenation of a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in $\R^n$.


Then the finite sequence of reversed directed smooth curves:

$-C_n, -C_{n - 1}, \ldots, -C_1$

defines a contour that is independent of the parameterization of $C_1, \ldots, C_n$.


Proof

Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.

From Reversed Directed Smooth Curve is Directed Smooth Curve, it follows that $-C_i$ is independent of the paraterization $\gamma_i$ of $C_i$.


We now prove that the end point of $-C_i$ is equal to the start point of $-C_{i - 1}$ for all $i \in \set {2, \ldots, n}$.

By definition of reversed directed smooth curve, $-C_i$ is parameterized by $\rho_i: \closedint {a_i} {b_i} \to \C$.

Here, $\map {\rho_i} t = \map {\gamma_i} {a_i + b_i - t}$.

From Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints, it follows that the endpoints $\map {\rho_i} {a_i}$ and $\map {\rho_i} {b_i}$ are independent of the parameterization $\rho_i$.

Then:

\(\ds \map {\rho_i} {b_i}\) \(=\) \(\ds \map {\phi_i} {a_i + b_i - b_i}\)
\(\ds \) \(=\) \(\ds \map {\phi_i} {a_i}\)
\(\ds \) \(=\) \(\ds \map {\phi_{i - 1} } {b_{i - 1} }\) Definition of Contour (Complex Plane)
\(\ds \) \(=\) \(\ds \map {\rho_{i - 1} } {a_{i - 1} }\)

By definition, it follows that $-C_n, \ldots, -C_1$ can be concatenated to form a contour.

$\blacksquare$


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