Length of Contour is Well-Defined

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Theorem

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

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

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


Suppose that $\sigma_i: \closedint {c_i} {d_i} \to \C$ is a reparameterization of $C_i$ for all $i \in \set {1, \ldots, n}$


Then:

$\ds \sum_{i \mathop = 1}^n \int_{a_i }^{b_i} \size {\map {\gamma_i'} t} \rd t = \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} \size {\map {\sigma_i'} t} \rd t$

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 $i \in \set {1, \ldots, n}$.

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

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

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

From Continuous Real Function is Darboux Integrable, we find that $\ds \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \size {\map {\gamma_i'} t} \rd t$ and $\ds \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} \size {\map {\sigma_i'} t} \rd t$ are defined.

Hence, all real integrals in the theorem are defined.


Then:

\(\ds \sum_{i \mathop = 1}^n \int_{a_i }^{b_i} \size {\map {\gamma_i'} t} \rd t\) \(=\) \(\ds \sum_{i \mathop = 1}^n \int_{\map {\phi_i^{-1} } {a_i} }^{\map {\phi_i^{-1} } {b_i} } \size {\map {\gamma_i'} {\map \phi u} } \map {\phi'} u \rd u\) substitution with $t = \map \phi u$
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \int_{\map {\phi_i^{-1} } {a_i} }^{\map {\phi_i^{-1} } {b_i} } \size {\map {\gamma_i'} {\map \phi u} \map {\phi'} u}\rd u\) as $\map {\phi'} u > 0$ by Derivative of Monotone Function
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \int_{\map {\phi_i^{-1} } {a_i} }^{\map {\phi_i^{-1} } {b_i} } \size {\map {\sigma'} u} \rd u\) Derivative of Complex Composite Function
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} \size {\map {\sigma'} u} \rd u\) Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints

$\blacksquare$