Given the two curves \begin{align*}&\mathcal{C}\left\{\begin{matrix}u = t\\v = t\end{matrix}\right., & t\in [0,1]\\ \\ &\mathcal{C'}\left\{\begin{matrix}u = t^3\\v = t^3\end{matrix}\right., & t\in [0,1]\end{align*} Show these to be equivalent.
Intuitively I see that they must represent the same line, but I fail to see how I can show this mathematically.
$t^3$ depends on $t$ as a direct function. Jacobian ( C, C') vanishes on its independent variables. By a function substitution both can be made identical.