Question

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

Answer 1

I think dA= dx dy
and they are saying
\[ \int \int du\ dv = \int \int | J(x,y) |\ dx\ dy \]

Answer 2

Oh, it makes perfect sense to me now. Thank you very much!