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I've been given the function $r(t) = ( \sqrt{2}t, e^t, e^{-t} ) $ and asked to find:

(a) the unit tangent and unit normal vectors, and

(b) the curvature using the formula |T'(t)| / |r'(t)| only (i.e. other formulae for K disallowed)

My answers for (a) match the textbook answer, but my answer for (b) does not. Unfortunately, the textbook only gives the final expected answer, no intermediate values or working, and I am unable to locate my error.

For reference I obtained:

r'(t) = (\sqrt{2}, e^t, -e^{-t} ) (unconfirmed)

$T(t) = \frac{1}{e^{2t} + 1}( \sqrt{2}e^t, e^{2t}, -1 )$ (apparently correct)

T'(t) = 2e^{2t} ( \frac{1 - e^{2t}}{\sqrt{2}e^t}, 1, 1 ) (unconfirmed)

$N(t) = \frac{1}{e^{2t} + 1} ( (1 - e^{2t}, \sqrt{2}e^t, \sqrt{2}e^t)$ (correct)

and K = |T'(t)| / |r'(t)| = \frac{(\sqrt{2}e^t)(1 + e^{2t})}{(e^{2t} + 1)/e^t} = \sqrt{2}e^{2t} (apparently incorrect)

However the text book obtains $K = \frac{\sqrt{2}e^{2t}}{(e^{2t} + 1)^2}$

Can anyone tell me where I've gone wrong?

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    @draks: Thanks so much for the links. I was very frustrated when posting this question as I didn't know how to control the formatting better, but these will definitely help me! Thanks for being constructive :)2012-02-25

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