Consider the mapping (u, v) ? (x(u, v), y(u, v)) defined by x(u,v) = 4u and y(u,v) = 2u + 3v. Let G denote the rectangle [0,1]

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himanshujoshi3738Lv5

21 Apr 2023

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Let G denote the region in the uv-plane given by a ? u ? b andg1(u) ? v ? g2(u) for two functions g1(u) and g2(u) such that g1(u)? g2(u) for u ? [a, b]. Consider the transformation x(u,v) = u andy(u,v) = ?(u,v) for some given function ? which is continuouslydifferentiable, and with the property that ??/?v is never zero.Next, assume that R, the image of G under this mapping (u, v) ???(u, ?(u, v)), is a similar region, meaning that R is given by a ? x? b and h1(x) ? y ? h2(x) for some functions h1(x) and h2(x)satsifying h1(x) ? h2(x) for x ? [a, b]. Show that if f : R ? R iscontinuous, then the double integral f(x,y)dxdy = the doubleintegral f(u,?(u,v))???v?? dudv.