Suppose $\alpha$ is a time dependent curve so that $\alpha:[0,T]\times I \to \mathbb{R} ^n$. I am a bit confused as to what the meaning of the expression $\partial_t(ds)$ is, where $ds = |\partial_x \alpha|dx$ is the arclength element, I am given.
How do I interpret it? Also how do I interpret $\partial_t(dx)$? Is this identically 0?
Thanks