- #1
kidsmoker
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Homework Statement
Let [tex]\underline{r}[/tex] be a regular parameterisation of a space curve [tex]C \subset R^{3}[/tex]. Prove that
[tex]\kappa=\frac{\left\|\underline{\dot{r}}\times\underline{\ddot{r}}\right\|}{\left\|\underline{\dot{r}}\right\|^{3}}[/tex] .
The Attempt at a Solution
We have
[tex]t(u)=\frac{\frac{dr}{du}}{\left\|\frac{dr}{du}\right\|}[/tex]
so differentiating both sides wrt u we obtain
[tex]\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\|})[/tex].
Since
[tex]\frac{dt}{du}=\kappa\underline{n}[/tex]
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!