I'm trying study the curve shortening flow for convex curves. I'm studying for the Lectures on Mean Curvature Flows by Xi-Ping Zhu and I found difficult to understand how the got the equation 1.2 (according the numeration of the book). He used two support functions to find equation 1.2, for the first function, he got $k = \frac{1}{S_{\theta \theta} + S}$ and, for the second function, he got $\frac{\partial S}{\partial t} = - k$, but he states that $\frac{\partial S}{\partial t} = - \frac{1}{S_{\theta \theta} + S}$. My doubt is why can he conclude it being that the two support functions are different? Next, the development to obtain the equation 1.2.
Thanks in advance!
Putting my comments from above in an answer as requested by the OP.
The two support functions $\langle\gamma, (\cos \theta,\sin \theta)\rangle$ and $\langle \gamma, -n \rangle$ are in fact the same (or correspond via reparametrization), since $\theta$ is the normal angle and thus $(\cos \theta, \sin \theta) = -n$ by definition.
The relation $k = d \theta/ds$ between the normal angle and the curvature is somewhat expanded upon in the last paragraph of this wikipedia section.