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A parametrized curve has a unit-speed reparametrization if and only if it is regular.

Proof: Suppose first that a parametrized curve γ: (a,b) → R^n has a unit-speed reparametrization μ, with reparametrization map φ. Letting t = φ(s) we have μ(s) = γ(t) and so

dμ/ds = (dγ/dt)(dt/ds)

Therefore, ||dμ/ds|| = ||dγ/dt|| ⋅ |dt/ds|

Since μ has unit speed, ||dμ/ds|| = 1, so dγ/dt cannot be zero.

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I don't see why it was necessary? Couldn't we have just concluded the exact same thing without the variable change? Another thing, I notice that they use absolute value for |dt/ds| this is because t = φ(s) is a single variable function right? So it's like a 'constant'? Was it just to make this more explicit? Because the norm properties don't have ||fg|| = ||f|| ||g||.

Thanks for the help!