Frenet frame in $\mathbb{R}^2$ My questions are probably very elementary ones, but I couldn't answer them for myself and didn't find the answer online.
Let $\gamma$ be a $C^2$ curve in $\mathbb{R}^2$ parametrized by arclength, and let $\{T(s),N(s)\} $ be it's frenet frame:


*

*Why is it true that $2\langle\dot{T}(s), T(s)\rangle = \frac{d}{ds} \langle T(s),T(s)\rangle$?

*What does $det[\gamma'(t),\gamma''(t)]$ mean? I don't understand this notation.


for the first
As symple said... Product rule
$$
\frac{d}{ds}\langle T(s), T(s)\rangle =
\langle T'(s), T(s)\rangle + \langle T(s), T'(s)\rangle
= 2 \langle T'(s), T(s)\rangle
$$  
for the second
Each of $\gamma'(t)$ and $\gamma''(t)$ is a $2$-dimensional vector.  Put them together into a $2 \times 2$ matrix, and take the determinant of that.  If this determinant is nonzero then $\gamma'(t)$ and $\gamma''(t)$ are not parallel ... And then the unit normal $N(s)$ can be defined.
