Length of Contour is Well-Defined

Theorem
Let $C_1, \ldots, C_n$ be directed smooth curves.

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

Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.

Suppose that $\sigma_i: \left[{c_i \,.\,.\, d_i}\right] \to \C$ is a reparameterization of $C_i$ for all $i \in \left\{ {1, \ldots, n}\right\}$

Then:


 * $\displaystyle \sum_{i \mathop = 1}^n \int_{a_i }^{b_i} \left\vert{ \gamma_i' \left({t}\right) }\right\vert \rd t = \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} \left\vert{ \sigma_i' \left({t}\right) }\right\vert \rd t$

and all real integrals in the equation are defined.

Proof
From the definition of directed smooth curve, it follows that $\sigma_i = \gamma_i \circ \phi_i$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

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

For all $i \in \left\{ {1, \ldots, n}\right\}$, $\gamma_i$ and $\sigma_i$ are continuous.

From Complex Modulus Function is Continuous and Continuity of Composite Mapping/Corollary, it follows that $\left\vert{ \gamma_i' }\right\vert$ and is $\left\vert{ \sigma_i' }\right\vert$ are continuous.

From Continuous Function is Riemann Integrable, we find that $\displaystyle \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \left\vert{ \gamma_i' \left({t}\right) }\right\vert \rd t$ and $\displaystyle \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} \left\vert{ \sigma_i' \left({t}\right) }\right\vert \rd t$ are defined.

Hence, all real integrals in the theorem are defined.

Then: