Let G denote the region in the uv-plane given by a ? u ? b and g1(u) ? v ? g2(u)

Question

Let G denote the region in the uv-plane given by a ? u ? b and g1(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 and y(u,v) = ?(u,v) for some given function ? which is continuously differentiable, 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 is continuous, then the double integral f(x,y)dxdy = the double integral f(u,?(u,v))???v?? dudv.

Explanation / Answer

Let G denote the region in the uv-plane given by a u b and g1(u) v g2(u) for two functions g1(u) and g2(u) such that g1(u) g2(u) for u 2 [a; b]. Consider the transformation x(u; v) = u and y(u; v) = (u; v) for some given function which is continuously dierentiable, and with the property that @ =@v is never zero. Next, assume that R, the image of G under this mapping (u; v) 7! (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 2 [a; b]. Show that if f : R ! R is continuous, then ZZ R f(x; y) dxdy = ZZ G f(u; (u; v)) @ @v dudv :

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