The question I am working on is the following: Show that $\alpha(s) = (5/13 \cos s, 8/13 - \sin s, -12/13 \cos s)$ is a unit speed curve.
So my working is:
$\alpha'(s) = (-5/13 \sin s, -\cos s, 12/13 \sin s)$
$| \alpha'(s)| = 1$.
But my textbook says that a curve $\alpha$ is a unit speed curve if $|d\alpha/dt| = 1$ not $| d\alpha/ds | = 1 $. I am really confused. What am I getting wrong?
For a vector-valued function $\alpha(\cdot)$ writing it as $t\mapsto\alpha(t)$ or as $s\mapsto\alpha(s)$ are mathematically the same thing. There is a slight semantical (or mathematic-sociological) difference: If you use the letter $s$ as variable you tacitly claim that $|\alpha'(s)|=1$, and if you use some other letter this tacit understanding is absent. In the case at hand it was appropriate to use the letter $s$, since your calculation shows that indeed $|\alpha'(s)|=1$.