Solution let s be the entire surface of the given

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Unformatted text preview: = (a2 − x2 − 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 S F · n dS = V · F dV (z 2 + x2 + y 2 ) dV . = V Copyright c 2014 The University of Sydney z 1 a x y 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 ≤ 2π, 0 ≤ ϕ ≤ π/2 . Now, x2 + y 2 + z 2 = r2 , and dV = r2 sin ϕ dr dθ dϕ. So, the flux = F · n dS S a 2π π /2 r4 sin ϕ dr dθ dϕ = 0 0 0 2πa5 . 5 = 3. Evaluate ( S × F) · n dS , where S is the surface of the hemisphere x2 + y 2 + z 2 = a2 for z ≥ 0, F = y i + zx j + y k and n is the unit normal with positive k-component. (Note that there are various ways you could do this. For example, you could calculat...
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