I have to evaluate the line integral of the vector field$$F(x,y)=\left(\frac{-2y}{x^2+y^{2}/4}+2x,\frac{2x}{x^{2}+y^{2}/4}+xy+1\right)$$through $\gamma$ where $\gamma$ is the ounterclockwise unit circle.
I just want to know if I can use Green's Theorem here; I don't remember if I could apply it directly due to the fact that, if $x=y=0$ then the denominator above is equal to zero.
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$\begingroup$Green's theorem requires that the function $F$ has partial derivatives which exist and are continuous over the entire interior of the interior of $\gamma$. Since the partial derivatives of $F$ do not exist at $(0,0)$, this means that the conditions required for Green's theorem do not hold.
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