Integrals of the form occurring here are called line or path integrals
over the line L. This equation follows from the same result in one dimensional
calculus when L is a straight line. But it holds just as well when L is a sufficiently
simple and smooth path in higher dimensional space. Such a path can be broken
up into tiny pieces, each one arbitrarily close to a straight line, so that
on each one the statement holds to arbitrary accuracy. Adding these statements
up implies the claim for the entire path.
This statement implies that the line integral of a gradient depends only
on the endpoints of integration and not on the details of the path chosen to
get from one to the other.