Question

Formulate and verify Green theroem when [P,Q]=[xy,1] and the curve (gama) is positively oriented circle containing the points:
A(0,0) ----> B(0,2) ----> C(1,1) ---> A(0,0)

Answer 1

any help?

Answer 2

so looks like you going positivly oriented around a triangle in the xy plane.
Lets call the boundary T
\[\int\limits_{∂T}^{}(xy)dx+1(dy)\] which according to brother green thats nothing more than
\[\int\limits_{T}^{}\int\limits_{}^{}∂(1)/∂x-∂(xy)/∂ydA\]
which is nothing more than\[\int\limits_{T}^{}\int\limits_{}^{}-xdxdy\]
then you set up your paramerization
A(0,0) ----> B(0,2) ----> C(1,1) ---> A(0,0)

now since you go clockwise in the A-->B-->C-->A
and in the drawing i go counterclockwise(positivly oriented)
just put a (-) in front of the whole integral
-∫∫(-x)dxdy=∫∫xdxdy
you may have to slipt it up into two different integrals