Week 14: Path independence, Potential functions and conservative fields, Green's theorem in the plane

Thomas Calculus, 11th edition, pages: 1160 - 1181

Sec 16.3 Path Idependence, Potential Functions and Conservative Fields

In gravitational and electric fields, the amount of work it takes to move a mass or a charge from one point to another depends only on the object’s initial and final positions and not on the path taken in between. This section discusses the notion of path independence of work integrals and describes the properties of fields in which work integrals are path independent. Work integrals are often easier to evaluate if they are path independent.

 Sec 16.4 Green's Theorem in the Plane

From Table 16.2 in Section 16.2, we know that every line integral  can be written as a flow integral If the integral is independent of path, so the field F is conservative (over a domain satisfying the basic assumptions), we can evaluate the integral easily from a potential function for the field. In this section we consider how to evaluate the integral if it is not associated with a conservative vector field, but is a flow or flux integral across a closed curve in the xy-plane. The means for doing so is a result
known as Green’s Theorem, which converts the line integral into a double integral over the region enclosed by the path.
We frame our discussion in terms of velocity fields of fluid flows because they are easy to picture. However, Green’s Theorem applies to any vector field satisfying certain mathematical conditions. It does not depend for its validity on the field’s having a particular physical interpretation.