User:Anghel/Sandbox

Theorem
Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves.

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

Define $\psi_i:\left[{a_i\,.\,.\,b_i}\right] \to \left[{a_i\,.\,.\,b_i}\right]$ by $\psi_i \left({t}\right) = a_i + b_i - t$.

Define $\rho_i:\left[{a_{n+1-i}\,.\,.\,b_{n+1-i}}\right] \to \C$ by $\rho_i = \gamma_{n+1-i} \circ \psi_{n+1-i}$.

Then $\rho_i$ is a smooth path which parameterizes a directed smooth curve $-C_i$.

The finite sequence $-C_1, \ldots, -C_n$ defines a contour $-C$ which is independent of the parameterization $\gamma_1, \ldots, \gamma_n$.

Proof
First, we prove that $\rho_i$ is a smooth path:

As $\gamma_{n+1-i}'$ and $\psi_{n+1-i}$ are continuous, it follows from Continuity of Composite Mapping that $\rho_i'$ is continuous.

Then $\rho_i$ is a parameterization of a directed smooth curve $-C_i$.

Next, we prove that the definition of $-C_i$ is independent of the parameterization $\gamma_{n+1-1}$.

Suppose $\sigma_i$ is another parameterization of $C_i$, so $\sigma_i = \gamma_i \circ \phi_i$.

Here, $\phi_i: \left[{c_i \,.\,.\, d_i}\right] \to \left[{a_i \,.\,.\, b_i}\right]$ is a bijective differentiable strictly increasing function.

Define $\tilde{\psi}_i:\left[{c_i\,.\,.\,d_i}\right] \to \left[{c_i\,.\,.\,d_i}\right]$ by $\tilde{\psi}_i \left({t}\right) = c_i + d_i - t$.

Define $\tilde{\rho}_i:\left[{c_i\,.\,.\,d_i}\right] \to \C$ by $\tilde{\rho}_i = \sigma_{n+1-i} \circ \tilde{\psi}_{n+1-i}$.

We now prove that $\rho_i$ and $\tilde{\rho}_i$ both are parameterizations of the same directed smooth curve $-C_i$.

Define $\tilde{\phi}_i:\left[{c_{n+1-i}\,.\,.\,d_{n+1-i}}\right] \to \left[{a_{n+1-i}\,.\,.\,b_{n+1-i}}\right]$ by $\tilde{\phi}_i = \psi_{n+1-i}^{-1} \circ \phi_{n+1-i} \circ \tilde{\psi}_{n+1-i}$.

From Composite of Bijections is Bijection, it follows that $\tilde{\phi}_i$ is bijective.