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The arclength of a parametrized segment (integration).

  1. May 6, 2012 #1
    1. The problem statement, all variables and given/known data
    For two points p and q in ℝ^n, use the formula (20.3) to check that the arclength of the parametrized segment from p to q is ||p - q||.


    2. Relevant equations
    Formula (20.3):
    A smooth parametrized path [tex]\gamma: [a, b]→ℝ^n[/tex]is rectifiable, and its arclength l is given by
    [tex]l = \int_{a}^{b}||\gamma '||[/tex]


    Norm of a point x = (x_1, ... , x_n) in ℝ^n is defined in this book as
    [tex]||x|| = \sqrt{x_1^2 + ... + x_n^2}[/tex]

    3. The attempt at a solution
    [tex]\gamma: [q, p]→ℝ^n[/tex]
    is defined as
    [tex]\gamma (t) = (\gamma _1 (t), \gamma _2 (t), ... , \gamma _n (t))[/tex]
    [tex]t \in [q, p][/tex]
    then
    [tex]\gamma '(t) = (\gamma _1 '(t), ... , \gamma _n '(t))[/tex]
    [tex]||\gamma '(t)|| = \sqrt{(\gamma _1 '(t))^2+ ... + (\gamma _n '(t))^2}[/tex]

    [tex]l = \int_{q}^{p} \sqrt{(\gamma _1 '(t))^2+ ... + (\gamma _n '(t))^2} dt [/tex]

    I can't figure out how to integrate this though to get to ||p - q||.
     
  2. jcsd
  3. May 6, 2012 #2

    LCKurtz

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    You need to write out the actual parameterization. Start with ##p=(p_1,p_2,...,p_n)## and ##q = (q_1,q_2,...q_n)##. Write out the parameterization of the line pq and work out the integral.
     
  4. May 6, 2012 #3
    I just realized p and q are in ##ℝ^n##. So [p, q] isn't an interval I can integrate over.
    I just found the equation I should probably use: ##\gamma:[0,1]→ℝ^n## defined as ##\gamma (t) = tq + 1(1 - t)p## for ##t \in [0, 1]##.
    Thanks for the hint.
     
  5. May 6, 2012 #4

    LCKurtz

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    Good start. Now calculate ##|\gamma'(t)|## and work out the ##t## integral. You can do it by components or work at this level.
     
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