please help (Green's theorem)?

Question

How can i verify the Green's theorem by using a computer algebra system to evaluate both the line integral and the double integral.
P(x,y) = x^4y^5, Q(x,y) = -x^7y^6, C is the circle x^2 + y^2 =1

Answer 1

to integrate P dx + Q dy, use a parametrization of the circle: (x(t),y(t)) = (cos(t),sin(t)).

Then you have
integral from 0 to 2pi [x(t)^4 y(t)^5 dx/dt -
x(t)^7 y(t)^6 dy/dt] dt

for the integral of dP/dx - dQ/dy over the unit disc, use polar coordinates: x(r,t) = r*cos(t), y(r,t) =r*sin(t). Then you get

integral 0 to 2pi integral 0 to 1 -7x(r,t)^6 y(r,t)^6 - 5x(r,t)^4 y(r,t)^4 r dr dt

You can use a computer algebra system such as Maple to evaluate these integrals to verify they're the same. In fact, I just used Maple to do this :)