I know $$\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \ dt = \int_C P dx + Q dy + Rdz $$
I understand the last two, since $dt$, and $dx, dy, dz$ are infinitesimal quantities of their variables. I'm trying to understand $d\vec{r} = \vec{r}'(t) \ dt$. Does this correspond to an "infinitesimal step" of $t$ along the curve $C$ that $r(t)$ traces out? And I presume $d\vec{S}$ is like an "infinitesimal rectangle", equal to $(\vec{r}_u \times \vec{r}_v) \,dA $? The notation is confusing to me.