Question

in a typical line integral how is it that the integral is expressed in terns of ds over a path C? wouldn't ds refer to a surface increment? not a path increment?

Answer 1

If you are doing a line integral you are doing it over a parameterized "curve" in space as opposed to a parameterized "surface". A curve can be parameterized by one independent parameter "t" and when you integral over it, you integrate over the differential arclength ds. However, when you integrate over a surface (in 3-D space), you have 2 independent variables "u" and "v". Then you integrate over a surface element dS.

Answer 2

ds is different than dS. malevolence19 is correct
It's a subtlety yes, but it's just something to keep in mind, especially in the context of a specific problem