**What is the importance of Green’s Theorem in a Plane (Boas, 3rd, Ed., Eqn. 9.7)?**

\int \int_A \left [ \frac{\partial Q}{\partial x} \frac{\partial P}{\partial y} \right ]\ dx\ dy = \int_C \left [ P\ dx + Q\ dy\right ] where P(x,y) and Q(x,y) are continuous functions with continuous first derivatives and C is the counterclockwise path that bounds area A.

This theorem relates an area integral involving the derivatives of two functions to a line integral involving just the two functions directly. It is ultimately based on the fundamental theorem of calculus. If the two functions happen to be two components of the same vector field, then you can derive both the divergence theorem and Stokes theorem. The work done by a conservative force is by definition path-independent. Stokes theorem then allows you to determine if a force is conservative without having to do any integrals. All you have to do to show that a force is conservative is to show that the curl of the force is zero. This is way easier than doing an infiinite number of integrals through every path imaginable and shows that all of these integrals give you the exact same result.