- #1
tuggler
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Given t\in Ithe arc length of a regular parametrized curve \alpha : I \to \mathbb{R}^3 from the point t_0 is by definition s(t) = \int^t_{t_0}|\alpha'(t)|dt where |\alpha'(t)| = \sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2} is the length of the vector \alpha'(t). Since \alpha'(t) \ne 0 the arc length s is a differentiable function of and ds/dt = |\alpha'(t)|.
This is where I get confused.
It can happen that the parameter tis already the arc length measured from some point. In this case, ds/dt = 1 =|\alpha'(t)|[/tex]. Conversely, if |\alpha&#039;(t)| = 1 then s = \int_{t_0}^t dt = t - t_0.<br /> <br /> How did they get that it equals 1? I am not sure what they are saying?
This is where I get confused.
It can happen that the parameter tis already the arc length measured from some point. In this case, ds/dt = 1 =|\alpha'(t)|[/tex]. Conversely, if |\alpha&#039;(t)| = 1 then s = \int_{t_0}^t dt = t - t_0.<br /> <br /> How did they get that it equals 1? I am not sure what they are saying?