Reparameterization of Directed Smooth Curve with Given Domain

Theorem
Let $\gamma: \closedint a b \to \C$ be a smooth path in the complex plane.

Let $C$ be a directed smooth curve with parameterization $\gamma$.

Let $\closedint { a_0 }{ b_0 }$ be a closed real interval, where $a_0 < b_0$.

Then there exists a smooth path


 * $\gamma_0 : \closedint { a_0 }{ b_0 } \to \C$

that is a reparameterization of $C$.

Proof
Define $\phi : \closedint a b \to \closedint { a_0 }{ b_0 }$ by:


 * $\map{ \phi }{ t } = \dfrac{ b_0 - a_0 }{ b - a } \paren{ t - a } + a_0$

Power Rule for Derivatives shows that


 * $\map{ \phi' }{ t } = \dfrac{ b_0 - a_0 }{ b - a }$

Real Function with Strictly Positive Derivative is Strictly Increasing shows that $\phi$ is strictly increasing, as $\map{ \phi' }{ t } > 0$ for all $t \in \closedint a b$.

Strictly Monotone Real Function is Bijective shows that $\phi$ is bijective.

Set $\gamma_0 = \gamma \circ \phi : \closedint a b \to \C$.

By definition of directed smooth curve, it follows that $\gamma_0$ is a reparameterization of $C$.