I am trying to find the Frenet frame of the following curve:
$$\zeta(t)=\left(\frac13(1+t)^{3/2},\frac{1}3(1-t)^{3/2},\frac{t}{\sqrt2}\right)$$
How do I do this? Is there a straightforward way from the curvature and Torsion? I can't find the definition anywhere.
Just look at any standard text on curves and surfaces, e.g., my own. As it happens, this curve is arclength-parametrized (i.e., $\|\zeta'(t)\|=1$ for all $-1<t<1$), and so it's just a matter of using the definitions — no chain rule needed.