# vcprac05s - The University of Sydney School of Mathematics...

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Unformatted text preview: The University of Sydney School of Mathematics and Statistics Solutions to Practice Session 5 MATH2061: Vector Calculus Summer School 2012 1. Evaluate the following surface integral: integraldisplayintegraldisplay S ( z + 2) dS, S : x 2 + y 2 + z 2 = a 2 . Solution: In spherical co-ordinates, the surface S is represented by x = a cos θ sin ϕ, y = a sin θ sin ϕ, z = a cos ϕ, (0 ≤ ϕ ≤ π, ≤ θ ≤ 2 π ) , and dS = a 2 sin φ dθ dφ. So we have integraldisplayintegraldisplay S ( z + 2) dS = integraldisplay 2 π integraldisplay π ( a cos ϕ + 2)( a 2 sin ϕ ) dϕ dθ = integraldisplay 2 π integraldisplay π ( a 3 sin ϕ cos ϕ + 2 a 2 sin ϕ ) dϕ dθ = integraldisplay 2 π bracketleftbigg a 3 sin 2 ϕ 2 − 2 a 2 cos ϕ bracketrightbigg π ϕ =0 dθ = 4 a 2 integraldisplay 2 π dθ = 8 πa 2 . 2. Find the flux of F = x i + y j + z k across the surface S , where S is the triangular region with vertices (1 , , 0) , (0 , 1 , 0) , (0 , , 1) . Solution: The surface S is the plane given by x + y + z = 1 ....
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vcprac05s - The University of Sydney School of Mathematics...

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