Arc length
Arc length of curves here is defined with respect to parametrisations - it is fundamental for defining unit speed parametrisations
Definition
TODO: Add picture
Like before we can take small steps [ilmath]\delta t[/ilmath] apart, the length of the line joining such points is [math]\|\gamma(t+\delta t)-\gamma(t)\|[/math] (where [math]\|\cdot\|[/math] denotes the Euclidean norm)
Noting that [math]\|\gamma(t+\delta t)-\gamma(t)\|\approx\|\dot{\gamma}(t)\delta t\|=\|\dot{\gamma}(t)\|\delta t[/math]
We can now sum over intervals, taking the limit of [math]\delta t\rightarrow 0[/math] we see that an infinitesimal section of arc length is [math]\|\dot{\gamma}(t)\|dt[/math].
Choosing a starting point [ilmath]t_0[/ilmath] we can define arc length, [ilmath]s(t)[/ilmath] as:
[math]s(t)=\int_{t_0}^t\|\dot{\gamma}(u)\|du[/math]
Rebasing arc length
Suppose we want the arc length to be measured from [ilmath]\widetilde{t_0} [/ilmath] then:
[math]\tilde{s}(t)=\int_{\widetilde{t_0}}^t\|\dot{\gamma}(u)\|du[/math] [math]=\int_{\widetilde{t_0}}^{t_0}\|\dot{\gamma}(u)\|du+\int_{t_0}^t\|\dot{\gamma}(u)\|du[/math] [math]=\int_{\widetilde{t_0}}^{t_0}\|\dot{\gamma}(u)\|du+s(t)[/math]
Differentiating arc length
Easy:
[math]\frac{d}{dt}\Big[s(t)\Big]=\frac{d}{dt}\Big[\int_{t_0}^t\|\dot{\gamma}(u)\|du\Big][/math][math]=\|\dot{\gamma}(t)\|[/math] by the Fundamental theorem of Calculus
Speed
Speed is the rate of change of distance (velocity is the rate of change of position - which are both vector quantities) - from differentiating the arc length (above) we define speed as:
The speed at [ilmath]t[/ilmath] of [ilmath]\gamma[/ilmath] is [math]\|\dot{\gamma}(t)\|[/math]
