Let $M$ be an interval on the real line or a smooth $1$-manifold diffeomorphic to $S^1$ and let $X:M\times[0,T]\to\mathbb{R}^2$ be a plane curve evolving according to $$\frac{\partial}{\partial t}X=-\phi N,$$ where $N$ is the unit normal and $\phi:M\times[0,T]\to\mathbb{R}$. If $\kappa$ is the curvature defined by $$\begin{cases} \frac{\mathrm{d}}{\mathrm{d}s}T=-\kappa N\\ \frac{\mathrm{d}}{\mathrm{d}s}N=\kappa T, \end{cases}$$ where $T$ is the unit tangent and $s$ is the arc length parameter, I'd like to show the arc length element $\mathrm{d}s$ satisfies $$\frac{\partial}{\partial t}\mathrm{d}s=-\phi\kappa\mathrm{d}s,$$ using $$\frac{\partial}{\partial t}\frac{\partial}{\partial s}=\phi\kappa\frac{\partial}{\partial s}.$$
Idea: If we interpret $\frac{\mathrm{d}}{\mathrm{d}s}$ as a vector field on $M$, then $\mathrm{d}s$ can be seen as a $1$-form dual to $\frac{\mathrm{d}}{\mathrm{d}s}$ that gives $$\mathrm{d}s\left(\frac{\mathrm{d}}{\mathrm{d}s}\right)=1.$$