Suppose we have the double integral of a function f(x,y) with domain of integration being some rectangular region in the 1st quadrant: 0≤a≤x≤b, 0≤c≤y≤d. Would the following transformation generally be acceptable? (I've quickly tried it out several times with arbitrary integrands and domains, with some instances in which I arrive at the correct answer, and others in which I do not; this leads me to believe that either it is a permissible change of variables but one has to be careful in certain cases, or it's generally not a permissible change of variables, hence the inconsistency in my answers.)(adsbygoogle = window.adsbygoogle || []).push({});

x=rtan(θ), y = rcot(θ), with 0≤[itex]\sqrt{ac}[/itex]≤r≤[itex]\sqrt{bd}[/itex], and 0≤arctan([itex]\sqrt{b/c}[/itex])≤θ≤arctan([itex]\sqrt{a/d}[/itex])≤π/2, s.t. we have dxdy -> |J|drdθ = 2r/cos(θ)sin(θ) drdθ.

I'm unsure of this transformation because although it fails to be one to one on the boundary of the region in the r-theta plane, it's one to one everywhere else, and the determinant only fails to be nonzero at r=0, which lies on the boundary of the said region in the r-theta plane as well.

Thanks in advance!

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