I'm having trouble proving that length is preserved under equivalent curves.
Equivalent curves are defined as such:
Two non-closed piecewise smooth parametrized curves are equivalent $\phi : [a, b] \rightarrow \mathbb{R}^n$ and $\psi : [c, d] \rightarrow \mathbb{R}^n$ are equivalent if there exists a continuous, piecewise $C^1$, strictly monotonic function $\gamma : [a, b] \rightarrow [c, d]$ such that $\gamma(a) = c$ and $\gamma(b) = d$ and $(\psi \circ \gamma )(t) = \phi(t)$ for all $t \in [a, b]$.
I'm trying to show that $\ell(\phi) = \ell(\psi)$. I know that $\ell(\phi) = \int_a^b \lVert(\phi)'(t)\rVert dt$ and that $\ell(\psi) = \int_a^b \lVert(\psi \circ \gamma)'(t)\rVert dt = \int_a^b \lVert(\psi'(\gamma(t)) \cdot \gamma'(t)\rVert dt$. But it seems as though I've hit a dead end.
Any help would be appreciated.