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Given the Field $F(x,y,z)=(y,z,x)$ and the curve $C$ that is the intersection of the plane $x+y=2$ and the sphere $(x-1)^2+(y-1)^2+z^2=2$. Evaluate $\oint_C F$ using Stokes' Theorem.

Ok, I firstly evaluate the curl. I know what's the $n$ vector because it is being cut by a plane whose equation was given. But my doubts are: I have to use this normal vector I know by the equation, I have to use this one in the equation or there are other ways?

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The given field ${\bf F}$ is a linear function, hence its curl ${\bf c}$ is a constant vector. We are going to apply Stokes' theorem to the disc $D$ cut out by the plane $x+y=2$ from the given sphere. The unit normal of $D$ is the vector ${\bf n}={1\over\sqrt{2}}(1,1,0)$. Compute the radius $R$ of $D$. Then, up to sign, $$\int_{\partial D}{\bf F}\cdot d{\bf r}=\int_D{\bf c}\cdot{\bf n}\>{\rm d}\omega={\bf c}\cdot{\bf n}\>\pi R^2\ .$$

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