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Problem: Use Green’s Theorem to prove the change of variables formula for a double integral

âˆ¬(f(x,y) dA) = âˆ¬(f(x(u,v),y(u,v)) |J(u,v)| du dv

for the case where f(x,y) is a function of x and y.

Here, R is the region in the xy-plane that corresponds to the region in the uv-plane under the transformation given by x = x(u,v) and y = y(u,v).

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Transcript

Hello, everyone in this problem. We are going to use the green theorem, so we know the green theorems integral over c m into dx plus n into dy to be equal to double integral over r. Do n by do x, minus to m. By do y of d x d y in the x y system, so if don by do x, minus do m by do y is f of x y and its value is 1. Then m d x, plus n d y will be equal to f of d x d y now. What is say so now here double integral over r d x d y, which is equal to integral c f of d x d y. So now considered the transformation x as x, of u v and y as y of u v. So then, the jacobin of this transformation is j of x y divided by u v to be equal to x, u x and y, u y…

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