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Homework Help: Unit Normal Vector N(t)

  1. Dec 7, 2007 #1
    1. The problem statement, all variables and given/known data
    This question comes from my multivariable differential calculus course, and it pertains to arc length parametrizations the unit normal vector of a curve.

    Why does the arc length parametrization of the unit normal vector of a curve, N(s) equal...

    _r''(s)__ = N(s) and not just equivalent to r''(s) = N(s)?
    ||r''(s)||

    2. Relevant equations
    T(s) = r'(s)
    ||r'(s)|| = 1



    3. The attempt at a solution
    I thought that since ||r'(s)|| = 1, ||r''(s)|| would be equivalent to 1 as well since they are both the normalizations of arc length parametrizations of curves. However, this apparently isn't the case...any help would be appreciated

    thanks
     
  2. jcsd
  3. Dec 7, 2007 #2

    Dick

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    Well, take a 'fer instance'. r(s)=(1/2)*(cos(2s),sin(2s)). r'(s) is unit length, r''(s) isn't.
     
  4. Dec 7, 2007 #3

    HallsofIvy

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    I'm just seconding Dick. If a curve is paraemtrised by arclength [itex]\vec{r}(s)[/itex], then it follows from the definition of "arclength", and the chain rule, that the length of [itex]\vec{r}'[/itex] is 1. There is no reason to expect that to be true for [itex]\vec{r}''[/itex] as well.

    Of course, [itex]\vec{r}''[/itex] is normal to the curve. I just doesn't have length 1.
     
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