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 $C$.
$\blacksquare$
Sources
- 2001: Christian Berg: Kompleks funktionsteori $\S 2.2$