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# SpSolHw10 - MATH-4600 Homework X Solutions to Graded...

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MATH-4600 Homework X Solutions to Graded Problems 1. 7.3 #6 F = ( x, y, z ) , D = { ( x, y, z ) : 0 z 9 - x 2 - y 2 } Gauss’s Theorem: S F · d S = V ∇ · F dV (a) -→ x ( s, t ) = ( s cos( t ) , s sin( t ) , 0) S F · d S = S bottom F · d S + S top F · d S S F · d S = 2 π 0 3 0 ( s cos( t ) , s sin( t ) , 0) · (0 , 0 , - s ) dsdt + 2 π 0 3 0 ( 2 s 3 cos 2 ( t ) + 2 s 3 sin 2 ( t ) + 9 s - s 3 ) dsdt = 2 π 3 0 ( s 3 + 9 s ) ds = 2 π s 4 4 + 9 s 2 2 3 0 = 243 π 2 (b) ∇ · F = 1 + 1 + 1 = 3 V ∇ · F dV = 3 V dV = 3 2 π 0 3 0 9 - r 2 0 rdzdθdr 6 π 3 0 ( 9 r - r 3 ) dr = 6 π 9 r 2 2 - r 4 4 3 0 = 243 π 2 2. 7.3 #7 F = ( y - x, y - z, x - y ) , D : 0 x, y, z 1 Gauss’s Theorem: S F · d S = V ∇ · F dV (a) S F · d S = S back F · d S + S front F · d S + S left F · d S + S right F · d S + S bottom F · d S + S top F · d S side x, y, z F ( x, y, z ) d S F · d S back x = 0 , 0 y, z 1 ( y, y - z, - y ) - i dA - ydydz front x = 1 , 0 y, z 1 ( y - 1 , y - z, 1 - y ) i dA ( y - 1) dydz left y = 0 , 0 x, z 1 ( - x, - z, x ) - j dA zdxdz right y = 1 , 0 x, z 1 (1 - x, 1 - z, x - 1) j dA (1 - z ) dxdz bottom z = 0 , 0 x, y 1 ( y - x, y, x - y ) - k dA - ( x - y ) dxdy top z = 1 , 0 x, y 1 ( y - x, y - 1 , x - y ) k dA ( x - y ) dxdy 1

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S F · d S = 1 0 1 0 - ydydz + 1 0 ( y - 1) dydz
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SpSolHw10 - MATH-4600 Homework X Solutions to Graded...

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