Arc length parametrisation question (error in notes?)

[chipotleaway]

This is from my course notes

http://img28.imageshack.us/img28/2630/ckyl.jpg

In line 3, there's the integral $$\int_0^t ||y'(s)||ds$$ which represents the length of the curve as a function of t (which I am thinking of as time). Here, I think s is a dummy variable for time.

The equation in line 4, however, says that s is an element of [0, |C|] which seems to imply that s iis a length. Here, it makes sense, because $\sigma$ maps time to length, and so $\sigma^{-1}$ maps length back to time, which goes into the function y and y(t) is spit out (the position vector function of the curve.

But back to the integrand, ||y'(s)|| where the variable s is a length makes no sense to me because y is meant to be function of time, not length. Is there something I'm misunderstanding here?

Thanks

 

[UltrafastPED]

t is the label on the x-axis; s is the arc length. The definitions used here have s=0 when t=0, and that s=|C| when t=1.

 

[chipotleaway]

Isn't t a time parameter which is not represented on the coordinate axes?

 

[lavinia]

- if you integrate the speed vector of a parameterized curve whose derivative is never zero( which guarantees that the curve can not back up on itself) the the integral equals the length of the curve.

- y is just the position in Euclidean space of the curve. It can be a function of many different parameters. s is the parameter whose value equals the length of the curve that has been traversed up to that point.

- There is no real concept of time here as in physics. t is just a parameter. s is another parameter.