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# reviewsol - 1(20 points Find the total ux upward throught...

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1. (20 points) Find the total flux upward throught the upper hemisphere ( z 0) of the sphere x 2 + y 2 + z 2 = a 2 of the vector field T ( x, y, z ) = x 3 3 i + yz 2 + e zx j + ( zy 2 + y + 2 + sin( x 3 )) k . Hint Use the divergence theorem around some ”simple” region. you’ll also need to compute one more flux integral. Use symmetry. Consider the solid region E defined by 0 z a 2 - x 2 - y 2 . Then ∂E has boundary S 1 = the upper hemisphere ( z 0) of the sphere x 2 + y 2 + z 2 = a 2 union S 2 = { ( x, y, z ) | z = 0& x 2 + y 2 = a 2 } . By the divergence theorem we get S 1 F · d S = E · F dV - S 2 F · d S . · F = x 2 + y 2 + z 2 so using spherical coordinates we get E · F dV = 2 π 0 π 2 0 a 0 ρ 2 ( ρ 2 sin φ ) dρdφdθ = 2 πa 5 5 . Next we need S 2 F · d S . On S 2 , ˆ n = - k so F · ˆ n = - (2 zy 2 + y + 2 + sin x 3 ). Since S 2 is planar, we get S 2 F · d S = x 2 + y 2 a 2 - (2(0) y 2 + y + 2 + sin x 3 ) dA xy = x 2 + y 2 a 2 - 2 dA xy (by symmetry) = - 2 πa 2 2. (20 points) Suppose a vector field is defined by F ( x, y, z ) = ( y 2 z ) i + (2 xyz ) j + ( xy 2 + 4 z ) k . a) Determine whether there is a scalar function φ ( x, y, z ) defined everywhere in space such that φ = F . If there is such a φ , find one; if there are none, explain why not. b) Determine whether there is a vector field A ( x, y, z ) defined everywhere in space such that × A = F . If there is such an A , find one; if there are none, explain why not.

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reviewsol - 1(20 points Find the total ux upward throught...

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