Consider the family of curves $$\gamma:S^1\times [0, T)\rightarrow \mathbb R^2, \gamma(u, t)=(x(u, t), y(u, t)),$$ where $u$ parametrizes the trace of the curve and $t$ parametrizes which curve is being considered. Can anyone explain me why I can write the system $$\frac{\partial\gamma}{\partial t}(u, t)=k(u, t)N(u, t),$$ as $$\frac{\partial}{\partial t}\left[\begin{array}{c} x\\ y\end{array}\right]=\frac{1}{\sqrt{x_u^2+y_u^2}}\left[\begin{array}{cc}y_u^2&-x_uy_u\\ -x_uy_u&x_u^2 \end{array}\right]\left[\begin{array}{c} x_{uu}\\ y_{uu}\end{array}\right].$$ Here $k(u, t)$ is the curvature and $N(u, t)$ is the unit inner normal vector.

A curve $\gamma$ satisfying the considered equation with an additional initial value is said to be evoluting by the curve shortening flow.

I need to write that equation in the matrix form for discussing the parabolic nature of the flow so that I could apply the parabolic theory of PDE for assuring short time existence for it..


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