It's been a while since I have done formal calculus and I am getting lost in a small detail when studying the first fundamental form of a regular surface $S$.
I am seeing this formulation in a set of notes I am reading: $$ \begin{align} ds = \left|\frac{d{\bf r}}{dt}\right|dt = \left|{\bf r}_u\frac{du}{dt}+{\bf r}_v\frac{dv}{dt}\right|dt &=\sqrt{({\bf r}_{u}\dot{u}+{\bf r}_{v}\dot{v})\cdot({\bf r}_{u}\dot{u}+{\bf r}_{v}\dot{v})}\,dt\\ &= \sqrt{Edu^2 + 2Fdudv + Gdv^2}\tag{3.11} \end{align} $$
I understand the intent, and I understand how/why the tangent to the curve is equal to the sum of the surface tangents as it is merely the chain rule. What I can't seem to recall is how and why the $dt$ pops up outside of the equation. Like $\frac{dr(t)}{dt}$ is trivially the tangent vector, and thus $\left|\frac{dr(t)}{dt}\right|$ is the magnitude, where is that second $dt$ term coming from?