reviewsol - 1. (20 points) Find the total flux upward...

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Unformatted text preview: 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. youll also need to compute one more flux integral. Use symmetry. Consider the solid region E defined by 0 z p 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 RR S 1 F d S = RRR E 5 F dV- RR S 2 F d S . 5 F = x 2 + y 2 + z 2 so using spherical coordinates we get RRR E 5 F dV = R 2 R 2 R a 2 ( 2 sin ) ddd = 2 a 5 5 . Next we need RR S 2 F d S . On S 2 , n =- k so F n =- (2 zy 2 + y + 2 + sin x 3 ). Since S 2 is planar, we get RR S 2 F d S = RR x 2 + y 2 a 2- (2(0) y 2 + y + 2 + sin x 3 ) dA xy = RR x 2 + y 2 a 2- 2 dA xy (by symmetry) =- 2 a 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 ( x,y, z ) defined everywhere in space such that 5 = F . If there is such a , find one; if there are none, explain why not....
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This note was uploaded on 10/18/2011 for the course MATH S1202Q taught by Professor Peters during the Spring '11 term at Columbia.

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reviewsol - 1. (20 points) Find the total flux upward...

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