Is $\int_C P dx + Q dy = \int_C \vec{F}\cdot d\vec{r}$?

Question

I am a bit confused about some notation. For a vector field $\vec{F}$, I understand the notation $$ \int_C \vec{F}\cdot d\vec{r}. $$ But I have also seen the notation $$ \int_C P dx + Q dy $$ If $\vec{F} = \langle P, Q\rangle$, are these two the same? (I am guessing that this is so from examples I have seen in calculating them.)

Answer 1

Yes, they are. If $\vec{F} = (P,Q)$ and $d\vec{r} = (dx,dy)$, then $$\vec{F} \cdot d\vec{r} = (P,Q) \cdot (dx,dy) = Pdx + Qdy$$