Hi William,
For a scalar field there are two possible ways to do the path integral. For each small step dl along the path, the first way sums up [scalar times dl], keeping track of the vector nature of dl. The second way simply sums up [scalar times length of dl] for each step. I believe in all cases the answer does depend on the path taken from point a to point b, with one exception: the first way, when the scalar field is constant everywhere. Then the integration is independent of path and proportional to the vector displacement b - a, and the integral around a closed loop = 0. But that case is not very interesting.
