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Calc III - Frénet-frame

  1. Mar 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the Frénet-frame of the streamline [itex]\textbf{r}(t) = \left(\frac{1}{2} \cosh t, e^t, \frac{1}{2} \cosh t\right)[/itex] at the point [itex](1,1,1)[/itex]

    2. Relevant equations

    [itex]\textbf{T}(t) = \frac{\textbf{r}'(t)}{||\textbf{r}'||}[/itex]
    [itex]\textbf{B}(t) = \frac{\textbf{r}'(t) \times \textbf{r}''(t)}{||\textbf{r}'(t) \times \textbf{r}''(t)||}[/itex]
    [itex]\textbf{N}(t) = \textbf{B}(t) \times \textbf{T}(t)[/itex]

    3. The attempt at a solution
    This is pretty straightforward. The only thing that is confusing me is what to do with [itex](1,1,1)[/itex]. Do I find T,B,N and plug [itex](1,1,1)[/itex] into that?

    Thanks
     
  2. jcsd
  3. Mar 4, 2012 #2

    Office_Shredder

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    Pretty much
     
  4. Mar 4, 2012 #3
    To be on the safe side here is how I calculated T.

    [itex]\textbf{r}'(t) = \left(\frac{1}{2} \sinh t, e^t, \frac{1}{2} \sinh t\right)[/itex]

    [itex]||\textbf{r}'(t)|| = \displaystyle \sqrt{(\frac{1}{2} \sinh t)^2 + (e^t)^2 + (\frac{1}{2} \sinh t)^2} = \sqrt{\frac{1}{2} \sinh ^2 t + e^{2t}} [/itex]

    So

    T(t) = [itex]\displaystyle \frac{\left(\frac{1}{2} \sinh t, e^t, \frac{1}{2} \sinh t\right)}{\sqrt{\frac{1}{2} \sinh ^2 t + e^{2t}}}[/itex]

    and

    T(1,1,1) = [itex]\displaystyle \frac{\left(\frac{1}{2} \sinh 1, e, \frac{1}{2} \sinh 1\right)}{\sqrt{\frac{1}{2} \sinh ^2 1 + e^{2}}}[/itex]
     
  5. Mar 4, 2012 #4

    Dick

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    Looks fine to me, so far.
     
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