# Line integral.

A path of integration is a parameterized plane curve , where the functions and are continuous and have continuous first derivatives for . We have:

 and

Let be a function continuous at every point of .

Example 5.1.2   Let be given by , where , i.e. is the upper half unit-circle.

Then and . we have:

Example 5.1.3   Let us compute the integral of the function along two different paths.

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).

Proposition 5.1.4

1. The integral is independent of the choice of the parameterization.
2. .
3. , when and are two paths such that the endpoint of and the origin of are identical.
4. If and are opposite paths, .

Noah Dana-Picard 2007-12-24