Let v = (x²z, 1 – 2xyz – 3y + x²y, 3z - x²z) be the velocity field of a fluid. Compute the flux of v across the surface x? + y? + z² = 4 where y > 0 and the surface is oriented away from the origin. HINT: Call the surface in this problem S1. Sı is "open" and does not enclose a 3D region, so Divergence Theorem cannot be used directly to calculate the flux across S1. Instead, try "capping" the S1 with a disk S2. Then the surface formed by combining S1 and S2 is a "closed" surfce S which does enclose a 3D region. Use the fact that F. dS = F· dS + F· dS and calculate F· dS by instead calculating F. dS (using Divergence Theorem) and calculating S1 S F· dS (using the original formula).

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