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?