Line Integration: Electric Field, Potential Energy & Displacement Vector

[Gear300]

I'm not exactly sure of this definition, but this seems to be it:
Line integration is the integration of a field...meaning the path that the field takes shape of and not the path of the object moving in the field. In that sense, the outcome of this integration only depends on the position the particle has at various points in the field.

Why is it that when integrating the Electric field in terms of electric potential and potential energy, that the displacement vector ds is taken to be a Scalar product with the Electric Field?

 

[blochwave]

Well that's from the very definition of a potential function. You're looking for a scalar function phi such that E=-grad(phi)

Then from the definition of the gradient and knowing calculus you can fish out that line integral that defines phi generally

 

[Gear300]

oh wait...thanks for reminding me. I'm beginning to understand how it works.