What is the Frénet-frame of a streamline at a given point?

[Jonmundsson]

Homework Statement

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]

Homework 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]

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

 

[Office_Shredder]

Pretty much

 

[Jonmundsson]

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]

 

[Dick]

Looks fine to me, so far.