Let be a function continuous at every point of .
Then and . we have:
First let be the same path as in previous example. If , then . and we have:
Now we compute the integral of along the segment from to on the axis; a parametrization of the segment is given by , for . Thus we have:
Note that the two paths have the same origin and the same endpoint, but the integrals are different. We will understand this phenomenon later (v.i. Corollary 3.4).
Noah Dana-Picard 2007-12-24