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

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Unformatted text preview: 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: I I S F · d S = Z Z Z V ∇ · F dV (a)-→ x ( s, t ) = ( s cos( t ) , s sin( t ) , 0) I I S F · d S = Z Z S bottom F · d S + Z Z S top F · d S I I S F · d S = Z 2 π Z 3 ( s cos( t ) , s sin( t ) , 0) · (0 , ,- s ) dsdt + Z 2 π Z 3 ( 2 s 3 cos 2 ( t ) + 2 s 3 sin 2 ( t ) + 9 s- s 3 ) dsdt = 2 π Z 3 ( s 3 + 9 s ) ds = 2 π s 4 4 + 9 s 2 2 3 = 243 π 2 (b) ∇ · F = 1 + 1 + 1 = 3 Z Z Z V ∇ · F dV = 3 Z Z Z V dV = 3 Z 2 π Z 3 Z 9- r 2 rdzdθdr 6 π Z 3 ( 9 r- r 3 ) dr = 6 π 9 r 2 2- r 4 4 3 = 243 π 2 2. 7.3 #7 F = ( y- x, y- z, x- y ) , D : 0 ≤ x, y, z ≤ 1 Gauss’s Theorem: I I S F · d S = Z Z Z V ∇ · F dV (a) I I S F · d S = Z Z S back F · d S + Z Z S front F · d S + Z Z S left F · d S + Z Z S right F · d S + Z Z S bottom F · d S + Z Z S top F · d S side x, y, z F ( x, y, z ) d S F · d S back x = 0 , ≤ y, z ≤ 1...
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This note was uploaded on 04/07/2008 for the course MATH 4600 taught by Professor Boudjelhka during the Spring '08 term at Rensselaer Polytechnic Institute.

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

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