Use the Divergence Theorem to evaluate  F · dS,  where  F(x, y, z) = z2xi +  + ((1/3)y3 + cos z)j + (x2z + y2)k  and S is the top half of the sphere  x2 + y2 + z2 = 4.  (Hint: Note that S is not a closed surface. First compute integrals over S1 and S2, where S1 is the disk  x2 + y2 ≤ 4,  oriented downward, and S2 = S1 ∪ S.)

Question

Use the Divergence Theorem to evaluate 

F · dS,

 where 

F(x, y, z) = z2xi +  + ((1/3)y3 + cos z)j + (x2z + y2)k

 and S is the top half of the sphere 

x2 + y2 + z2 = 4.

 (Hint: Note that S is not a closed surface. First compute integrals over S1 and S2, where S1 is the disk 

x2 + y2 ≤ 4,

 oriented downward, and S2 = S1 ∪ S.)

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