reparametrization function

Question

I am trying to understand some computation, but it seems something is not going well.

You will find this slide in the following link:

https://www.asc.ohio-state.edu/kurtek.1/Lecture3_Srivastava.pdf (no. 28)

For me it is not clear from where we have: $$(( r\circ \gamma)\dot{\gamma}, \theta \circ \gamma)?$$

Answer 1

Differentiate $f\circ \gamma$ (I use primes instead of overdots):

$$(f\circ \gamma)'=[f( \gamma(t))]'=f'( \gamma)\gamma'(t)=(f'\circ \gamma)\gamma'.$$ Thus, we have that $$|(f\circ \gamma)'|=|f'( \gamma)||\gamma'(t)|=(r\circ\gamma)|\gamma'|=(r\circ\gamma)\gamma',$$ if $\gamma'\ge 0.$

Also, we have that $$\frac{(f\circ \gamma)'}{|(f\circ \gamma)'|}=\frac{(f'\circ \gamma)\gamma'}{(r\circ\gamma)\gamma'}=\frac{f'(\gamma)}{r(\gamma)}=\Theta\circ \gamma.$$