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Follow-up to this: Does "$s'(t_0) = ||c'(t_0)||$" actually mean "$s'(t_0) \cong ||c'(t_0)||$" or "$\dot s(t_0) = ||c'(t_0)||$"?

Given "$s'(t_0) = ||c'(t_0)||$" actually means $\dot s(t_0) = ||c'(t_0)||$ (or "$s'(t_0) \cong ||c'(t_0)||$"), does "$\gamma'(s) = c'(t(s))t'(s)$" actually mean "$\gamma'(s) = c'(t(s))\dot t(s)$" (or "$\gamma'(s) \cong c'(t(s)) t'(s)$")?

It appears both $\gamma'(s)$ and $c'(t(s))$ are tangent vectors in the same tangent space, but $t'(s)$ is a tangent vector in a different tangent space.

William Elliot
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    This is where it gets ridiculous, but your first correction is again correct, $\dot t(s)$ is a proportionality factor between two tangent vectors. The second resp. original right side makes no sense, you would have to specify what product of tangent vectors (from different vector spaces) to use, even if $V\otimes_{\Bbb R}\Bbb R\simeq V$ for real vector spaces $V$. – Lutz Lehmann Jul 27 '19 at 11:57
  • @LutzL Thanks! You can answer (again) if you want. –  Jul 27 '19 at 12:01

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