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A biconditional statement for arc length of a function

  1. Mar 26, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that [itex]\gamma : [a, b] \rightarrow \Re^{2}[/itex] is a parameterization of [itex]\Gamma[/itex] if and only if the length of the curve from [itex]\gamma(a)[/itex] to [itex]\gamma(s)[/itex] is [itex]s - a[/itex]; i.e.,

    [itex]

    \int ^{s}_{a} \left| \gamma ' (t) \right| dt = s - a.

    [/itex]

    2. Relevant equations



    3. The attempt at a solution

    Part 1; show [itex]\left| \gamma ' (s) \right| = 1 \Rightarrow \int ^{s}_{a} \left| \gamma ' (t) \right| dt = s - a.[/itex]

    We have

    [itex]

    \ell = \int ^{b}_{a} \left| \gamma ' (t) \right| dt = \int ^{b}_{a} \left| \gamma ' (s(t)) s'(t) \right| dt = \int ^{b}_{a} \left| \gamma ' (s(t))\right| \left| s'(t) \right| dt = \int ^{\ell}_{0} \left| \gamma ' (s) \right| ds = \int ^{\ell}_{0} 1 ds = s(\ell) - s(0) = s - a.

    [/itex]

    My question here is: did I do the change of variables correctly? Specifically in the limits of integration?

    Now how do I prove the converse; i.e., how do I show that [itex]\int ^{s}_{a} \left| \gamma ' (t) \right| dt = s - a \Rightarrow \left| \gamma ' (s) \right| = 1 ?[/itex]

    [itex]\int ^{s}_{a} \left| \gamma ' (t) \right| dt [/itex]
    [itex]= \int ^{s}_{a} \sqrt{x'(t)^{2} + y'(t)^{2} } dt [/itex]
    [itex]= \int ^{s}_{a} \sqrt{ (x'(s(t)) s'(t) )^{2} + (y'(s(t)) s'(t) )^{2} } dt [/itex]
    [itex]= \int ^{s}_{a} \sqrt{ s'(t)^{2} (x'(s)^{2} + y'(s)^{2}) } dt [/itex]
    [itex]= \int ^{s}_{a} s'(t) \sqrt{ (x'(s)^{2} + y'(s)^{2}) } dt [/itex]
    [itex]= \int ^{s}_{t=a} \sqrt{ (x'(s)^{2} + y'(s)^{2}) } ds[/itex]
    [itex]= \int ^{s}_{t=a} \left| \gamma' (s) \right| ds = s - a[/itex]
    [itex]= \frac{d}{ds}(\int ^{s}_{t=a} \left| \gamma' (s) \right| ds)[/itex]
    [itex] = \frac{d}{ds}(s - a)[/itex]
    [itex] = 1 - 0 = 1[/itex]

    so [itex]1 = \left| \gamma' (s) \right| .[/itex]

    or is that garbage?
     
    Last edited: Mar 26, 2013
  2. jcsd
  3. Mar 26, 2013 #2
    This statement is incorrect. Is it supposed to be "natural parametrization" rather than just "parametrization?
     
  4. Mar 26, 2013 #3
    I do believe that the statement I am to prove is correct. I have copied it directly out of my text. My instructor says my proof is complete. I'm not sure what you meant.

    And it is natural parametrization.
     
  5. Mar 26, 2013 #4
    The statement is correct only if it is about natural parametrization. It is not correct for an arbitrary parametrization.
     
  6. Mar 26, 2013 #5
    Oh my bad. I failed to mention that in my
     
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