Use Greens Theorem to evaluate ∫F dr where F(x,y) = <y - ln(x^2 + y^2), 2arctan(y/x)>...?
Use Greens Theorem to evaluate ∫F dr where F(x,y) = <y - ln(x^2 + y^2), 2arctan(y/x)> and C is the circle (x-2)^2 + (y-3)^2 = 1 oriented counterclockwise.
Is there an easy way to do this because i got a huge mess of integrals when i tried it
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∫c F · dr
= ∫c [(y - ln(x^2 + y^2)) dx + 2 arctan(y/x) dy]
= ∫∫ [(∂/∂x) 2 arctan(y/x) - (∂/∂y)(y - ln(x^2 + y^2))] dA, by Green's Theorem
= ∫∫ [(2/(1 + (y/x)^2)) * (-y/x^2) - (1 - 2y/(x^2 + y^2))] dA
= ∫∫ [(-2y/(x^2 + y^2) - (1 - 2y/(x^2 + y^2))] dA
= ∫∫ 1 dA
= (Area enclosed by the circle (x-2)^2 + (y-3)^2 = 1)
= π.
I hope this helps!