I was told a proof that the electric field was conservative (without using $\nabla$) which used a point charge and showed the following:
$$w.d.=\int_c{\vec F \cdot \mathrm{d} \vec l}=\int_c{\vec F\cos(\theta) \mathrm{d} l}$$ where $c$ is a path from a to b and $\theta$ is the angle between $\vec F$ and $\mathrm{d}\vec l$. In the case of a point charge it can be shown that in the limit (i.e. the limit of small elements of path getting infinitesimal) we have $dr=cos(\theta)dl$ where $dr$ is the radial distance from the point charge. It is now this next step that I have problems with, every where I have seen this proof they do the following:
$$\int^{r_b}_{r_a}{F\cos(\theta)\mathrm{d}r}$$ This is no longer a line integral, why? (they then go onto say that this is path independent and there the force is conservative). So why have we suddenly jumped from a line integral to not a line integral. (a specific and general reason would be appreciated)
Here is a link to the proof.