\[\Large{\vec{F} = x(y-z)\hat{\imath} + y(z-x)\hat{\jmath} + z(x-y)\hat{k}}\]

Determine the surface integral of the vector field \(\vec{F}\) over a unit sphere centered on the origin.

**Note:** \(\hat{\imath}, \hat{\jmath},\hat{k}\) are unit vectors associated with the three Cartesian coordinate axes (\(x,y\), and \(z\) respectively).

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