Chain rule in parametrisation

Question

I couldn't understand the boxed highlighted step that uses chain rule in Proposition $1.3$. How is the chain rule applied in this case? Should't it be $t$ instead of $t$~ in the denominator?

Answer 1

Note that $t = \phi(\overset{\sim}{t})$. Hence, any derivative of $\phi$ will be taken with respect to the variable $\overset{\sim}{t}$. Therefore, the chain rule at a point $u$ will look like: $$ 1 = \frac{d(\phi \circ \psi)}{dt}(u) = \frac{d\phi}{d\overset{\sim}{t}}(\psi(u))\frac{d\psi}{dt}(u) $$

and this will be true for all $u$, so the above is succinctly written as: $$ 1 = \frac{d\phi}{d\overset{\sim}{t}}\frac{d\psi}{dt} $$

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