Question: Let \(\mathbf{F}=y\mathbf{i}-x\mathbf{j}+z\mathbf{k}\) and let \(f\left( x,y,z \right)=2x+2y+3z\) .

1. Let S1 be the part of the plane x + 2y + 3z = 1 in the first octant, with normal directed downwards. Compute both \(\int\limits_{{{S}_{1}}}{fdS}\) and \(\int\limits_{{{S}_{1}}}{\mathbf{F}\cdot dS}\)
2. Let S2 be the upper half of the surface of the sphere with center the origin and radius 1, with normal oriented upwards and S3 be the disk on the x, y-plane with center the origin and radius 1, with normal oriented downwards. Compute \(\int\limits_{{{S}_{2}}}{\mathbf{F}\cdot dS}+\int\limits_{{{S}_{3}}}{\mathbf{F}\cdot dS}\)

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