When we do a line integral over a scalar field, I understand this as a special case of an Integral over a one-dimensional real manifold. Getting the volume (in this case "length" as a 1-dim. volume) of the manifold involves using the gramian determinant. Now, an integral over a vectorfield should follow the same principle. But where does it make use of the gramian determinant?
What the method seems to do, is doing a projection of the vectorfield at that point onto the tangent space of the curve. And summing up all those projections lengths. Like in this animation. But maybe this flawed: a projection needs the vector we project onto to be of length 1. In this (infinitesimal) sum it is not.
