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quiz3-T-sol - Quiz 3 Tuesday(math 32b f(x y)dxdy in Problem...

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Quiz 3 - Tuesday (math 32b) November 14, 2007 Problem 1. For a function f ( x,y ) , rewrite the integral integraltext integraltext S f ( x,y ) dxdy in variables ( r,ϕ ) . Here the transformation ( r,ϕ ) mapsto→ ( x,y ) is given by x = r cos 3 ϕ , y = r sin 3 ϕ and S is the area bounded by the curve x 2 / 3 + y 2 / 3 = 1 . Solution. First, find the image of the region under the given transforma- tion: r 2 / 3 cos 2 ϕ + r 2 / 3 sin 2 ϕ = 1 r 2 / 3 = 1 r = 1 Thus, image is the disc of radius 1 centered at the origin. Second, compute the Jacobian: ( x,y ) ( u,v ) = det parenleftbigg cos 3 ϕ 3 r cos 2 ϕ sin ϕ sin 3 ϕ 3 r sin 2 ϕ cos ϕ parenrightbigg = = (3 r (sin ϕ 2 cos 4 ϕ + sin 4 ϕ cos 2 ϕ )) = = 3 r sin 2 ϕ cos 2 ϕ (cos 2 ϕ + sin 2 ϕ ) = 3 r sin 2 ϕ cos 2 ϕ. Thus, the integral is equivalent to
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