Let be a compact interval of , and is a smooth covector on .
Since we are working over a compact interval, has a smooth component function which can be extended to some neighborhood of the interval.
Let be the standard coordinate on , then .
We define the integral of over to be
in the normal calculus way.
For arbitrary manifolds
Let be a smooth manifold and .
If is a smooth curve on , we define the line integral of over to be
If is piecewise smooth, we can add all the integrals over the smooth components.
Computation using regular integrals
If is a piecewise smooth curve on , the line integral of over can be expressed as the integral
Fundamental Theorem for Line Integrals
Let be a smooth manifold.
For a smooth function , and a piecewise smooth curve we have