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Compute the curl?

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= (∂/∂x i + ∂/∂y j) X ((x-y)/(x^2+y^2) i,  (x+y)/(x^2+y^2) j)

= (x^2+y^2 - 2x(x+y))/(x^2+y^2)^2 k, - (x^2+y^2 - 2y(x-y))/(x^2+y^2)^2 k

= (-2x^2-2y^2)/(x^2+y^2)^2 k

= -2k/(x^2+y^2) <==Answer


Attn: If you are not familiar with this approach, you can use 3 by 3 determinant to get the same result.