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Differential Geometry, curve length

  1. Jan 14, 2012 #1
    1. The problem statement, all variables and given/known data

    paonb.png

    2. Relevant equations

    [itex]L[c]:=\int_{a}^{b}(\sum_{i,j=1}^{2}g_{ij}(c(t))c_{i}'(t)c_{j}'(t))^{1/2}dt[/itex]

    3. The attempt at a solution

    So [itex]g_{ij}(x,y)=0[/itex] for [itex]i{\neq}j[/itex], [itex]c_{1}'(t)=-Rsin(t)[/itex], [itex]c_{2}'(t)=Rcos(t)[/itex]

    so [itex]L[c]:=\int_{a}^{b}(\frac{1}{((Rsin(t))^{2}}R^{2}(sin^{2}(t)+cos^{2}(t))^{1/2}dt=\int_{a}^{b}\frac{1}{sin(t)}dt[/itex]

    However the solutions has

    [itex]L[c]:=\int_{a}^{b}(\frac{1}{((Rcos(t))^{2}}R^{2}(sin^{2}(t)+cos^{2}(t))^{1/2}dt=\int_{a}^{b}\frac{1}{cos(t)}dt[/itex]

    and he then goes on to use the given identity to find an antiderivative for [itex]\frac{1}{cost}[/itex]

    but I don't see how he has cost where I have sint.

    Is he making a mistake or am I?
     
    Last edited: Jan 14, 2012
  2. jcsd
  3. Jan 14, 2012 #2

    MathematicalPhysicist

    User Avatar
    Gold Member

    I wonder how come the hint is right but the solution in the text isn't.
     
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