Use the Divergence Theorem to evaluate F · dS, where F(x, y, z) = z2xi + + ((1/3)y3 + cos z)j + (x2z + y2)k andS is the top half of the spherex2 + y2 + z2 = 4. (Hint: Note thatS is not a closed surface. First compute integrals overS 1 andS 2, whereS 1 is the diskx2 + y2 ≤ 4, oriented downward, andS 2 =S 1 ∪S.)

Use the Divergence Theorem to evaluate F · d S , where F (x, y, z) = z 2 x i + + ((1/3)y 3 + cos z) j + (x 2 z + y 2 ) k and S is the top half of the sphere x 2 + y 2 + z 2 = 4. ( Hint : Note that S...

Use the Divergence Theorem to evaluate

F · dS,

where

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

andS is the top half of the sphere

x² + y² + z² = 4.

(Hint: Note thatS is not a closed surface. First compute integrals overS
₁ andS
₂, whereS
₁ is the disk

x² + y² ≤ 4,

oriented downward, andS
₂ =S
₁ ∪S.)

Jun 05, 2022