# How to justify the equality $-\int_{0}^{2\pi}k^2v\ du=-\int_{0}^Lk^2ds$ in this proof?

Consider the curvature flow $$\frac{\partial F}{\partial t}=kN,$$ where $k$ is the curvature and $N$ the inner unit normal and $F:S^1\times [0, T)\rightarrow \mathbb R^2$ is a family of closed curves. If $L$ is the arc length I have the following result $$\frac{\partial L}{\partial t}=-\int k^2\ ds.$$ The proof is as follows: $$\frac{\partial L}{\partial t}=\int_{0}^{2\pi}\frac{\partial}{\partial t}v\ du=-\int_{0}^{2\pi}k^2v\ du=-\int_{0}^Lk^2ds.$$ I don't know how to justify the last equality, what has been done? Observation: I also know that $\partial v/\partial t=-k^2v$.

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