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vcprac06s - The University of Sydney School of Mathematics...

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The University of Sydney School of Mathematics and Statistics Solutions to Practice Session 6 MATH2061: Vector Calculus Summer School 2014 1. Evaluate ii S F · n dS , where F = 2 x i +3 xy j yz 2 k and S is the surface bounded by the planes x = 0 , x + y = 2 , y = 0 , z = 0 and z = 2 . ( n is the unit outward normal to the surface S. ) Solution: Let V be the solid enclosed by S . Then V is described by: 0 x 2 0 y 2 x 0 z 2 z y x 2 2 2 Now, div F = ∇· F = 2 + 3 x 2 yz , and so by the divergence theorem, S F · n dS = iii V F dV = V (2 + 3 x 2 yz ) dV = i 2 0 i 2 0 i 2 - x 0 (2 + 3 x 2 yz ) dy dx dz = i 2 0 i 2 0 ((2 + 3 x )(2 x ) (2 x ) 2 z ) dx dz = i 2 0 b 4 x + 2 x 2 x 3 + z (2 x ) 3 3 B 2 0 dz = i 2 0 p 8 8 z 3 P dz = 32 3 . 2. Find the ±ux of F = xz 2 i +( x 2 y z 3 ) j +(2 xy + y 2 z ) k outwards across the entire surface of the hemispherical region bounded by z = ( a 2 x 2 y 2 ) 1 / 2 and z = 0 . Solution: Let S be the entire surface of the given region, and let V be the solid enclosed by S . Then y x z a S F · n dS = V F dV = V ( z 2 + x 2 + y 2 ) dV . Copyright c c 2014 The University of Sydney 1
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Using the spherical coordinates x = r cos θ sin ϕ , y = r sin θ sin ϕ , z = r cos ϕ , V is described by: 0 r a, 0 θ 2 π, 0 ϕ π/ 2 .
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