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quekyuxuan
Hi all, I'm looking at Part II of Problem Set 7, on Double Integrals. In the 4th paragraph of the background introduction to problems 4 and 5, it states "The general change-of-variables formula says that if a region R goes to a region R' by a transformation (x,y) → (X,Y) with Jacobian ∂(X,Y)/￼∂(x,y), then the areas of R and R' are related by A(R') = ∬|J(x,y)| dA." Shouldn't it be "A(R') = ∬|1/J(x,y)| dA" instead? For example, let h=h(x,y); u=x+y; v=x-y. |J(x,y)|=|∂(u,v)/￼∂(x,y)|=1-(-1)=2; dudv=2dydx ∬dydx = ∬1/|J(x,y)| dudv = ∬0.5 dudv Thank you
I think dA= dx dy and they are saying \[ \int \int du\ dv = \int \int | J(x,y) |\ dx\ dy \]
Oh, it makes perfect sense to me now. Thank you very much!