Change of variables Consider the parameterized curves r(t)

QUESTION:

Change of variables Consider the parameterized curves \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) and \(\mathbf{R}(t)=\langle f(u(t)), g(u(t)), h(u(t))\rangle\), where f,g, h, and u are continuously differentiable functions and u has an inverse on [a, b].

a. Show that the curve generated by r on the interval \(a \leq t \leq b\) is the same as the curve generated by \(\mathbf{R} \text { on } u^{-1}(a) \leq t \leq u^{-1}(b)\) \(\left(\text { or } u^{-1}(b) \leq t \leq u^{-1}(a)\right)\).

b. Show that the lengths of the two curves are equal. (Hint: Use the Chain Rule and a change of variables in the are length integral for the curve generated by R.)