How can we prove the curvature of a space curve using regular parameterization?

[kidsmoker]

Homework Statement

Let \underline{r} be a regular parameterisation of a space curve C \subset R^{3}. Prove that

\kappa=\frac{\left\|\underline{\dot{r}}\times\underline{\ddot{r}}\right\|}{\left\|\underline{\dot{r}}\right\|^{3}} .

The Attempt at a Solution

We have

t(u)=\frac{\frac{dr}{du}}{\left\|\frac{dr}{du}\right\|}

so differentiating both sides wrt u we obtain

\frac{dt}{du}=\frac{\frac{d^{2}r}{du^{2}}}{\left\|\frac{dr}{du}\right\|}+\frac{dr}{du}\frac{d}{du}(\frac{1}{\left\|\frac{dr}{du}\right\|}).

Since

\frac{dt}{du}=\kappa\underline{n}

this gets me the curavture in terms of the desired bits (with n too) but I can't seem to get it to the desired result :\

Thanks for your help!

 

[n!kofeyn]

This http://books.google.com/books?id=UXPyquQaO6EC&lpg=PP1&pg=PA25" is from a great introductory book to differential geometry. I highly recommend it.