Homework Help: Calc III - Frénet-frame

1. Mar 4, 2012

Jonmundsson

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

2. Relevant equations

$\textbf{T}(t) = \frac{\textbf{r}'(t)}{||\textbf{r}'||}$
$\textbf{B}(t) = \frac{\textbf{r}'(t) \times \textbf{r}''(t)}{||\textbf{r}'(t) \times \textbf{r}''(t)||}$
$\textbf{N}(t) = \textbf{B}(t) \times \textbf{T}(t)$

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

Thanks

2. Mar 4, 2012

Office_Shredder

Staff Emeritus
Pretty much

3. Mar 4, 2012

Jonmundsson

To be on the safe side here is how I calculated T.

$\textbf{r}'(t) = \left(\frac{1}{2} \sinh t, e^t, \frac{1}{2} \sinh t\right)$

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

So

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

and

T(1,1,1) = $\displaystyle \frac{\left(\frac{1}{2} \sinh 1, e, \frac{1}{2} \sinh 1\right)}{\sqrt{\frac{1}{2} \sinh ^2 1 + e^{2}}}$

4. Mar 4, 2012

Dick

Looks fine to me, so far.