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\noindent {\bf Problem statement} Let $C_1$ be the circle $x^2+y^2=16$
and $C_2$ the circle $(x-1)^2+y^2=4$, each oriented
counterclockwise. Suppose $\displaystyle P(x,y) = {x\over x^2+y^2}+y$
and $\displaystyle Q(x,y) = {y\over x^2+y^2}-x.$

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\noindent a) Verify that
 $\displaystyle{{\partial Q\over\partial x}(x,y)-{\partial P\over\partial y}(x,y) =
-2}$ for $(x,y)\ne(0,0)$. 
 
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\noindent b) Explain why Green's Theorem cannot be used to calculate
either $\displaystyle \int_{C_1}(P\,dx+Q\,dy)$ or $\displaystyle
\int_{C_2}(P\,dx+Q\,dy)$.
 
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\noindent c) Use Green's Theorem to show that $\displaystyle
\int_{C_1}(P\,dx+Q\,dy)=\int_{C_2}(P\,dx+Q\,dy)-24\pi$.










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