Understanding the arc length parameter notation

Question

I've been struggling with the notion of the arc length parameter and I was wondering if this was due to notations. In several text books I often see this occur. A regular curve $\sigma:[a,b]\rightarrow \mathbb{R}^n$ which is not parametrized according to arc length and so takes in as variable $t$ or rather has domain over $[a,b]$ the reparametrization of $\sigma$ is then $\sigma \circ s^{-1}$ where $s(t)= \int_{a}^{t}\lVert \sigma' (x)\rVert \,dx$ so $s:[a,b]\rightarrow[0,L(\sigma)]$. From what I understand a function that is evaluating the parameter $s$ has domain over $[0,L(\sigma)]$. But then there appears expression of the form $\sigma(s(t))$ which generally don't make sense Example:

My suspicion is that there is an abuse in notation where the reparametrization of $\sigma$ is just written as $\sigma$ is this the case. Also if this is the case is there a reason behind this use of notation?

Answer 1

In my experience, yes, it is typical for there to be abuse of notation when it comes to parametrization by arc length.

I think it typically comes with the use of a so-called "dummy variable" $t$ in the calculation of the integral $s(t)=\int_0^t||\sigma'(t)||\rm{dt}$.

I think it is often done in integrals of this type, and is just the nature of the beast.

Answer 2

The notation does not look too badly abused.

$\sigma' = \frac {d\sigma}{dt} = \frac {d\sigma}{ds}\frac{ds}{dt} = \dot\sigma\frac{ds}{dt}$ is just the chain rule and the given definitions.

$\frac{ds}{dt} = \|\sigma'(t)\|$ comes from the definition of $s$ and the fundamental theorem of calculus.