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Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi + y3 3 + cos z j + (x2z + y2)k and S...

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Question

Basic Math

Use the Divergence Theorem to evaluate

S

F · dS,

where

F(x, y,z) =z2xi+

y3
3

+ 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 andS2, where S1 is thedisk

x2 +y2 ≤ 4,

oriented downward, and S2 =S1 ∪ S.)

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