(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let U be the open set in [itex]R^2[/itex] consisting of all x with (Euclidean norm) [itex] ||x|| < 1.[/itex] Let [itex] f(x,y) = 1/(x^2 + y^2)[/itex] for [itex] (x,y) \not = 0.[/itex] Determine whether [itex] f [/itex] is integrable over [itex] R^2 - \overline{U}; [/itex] if so, evaluate it.

2. Relevant equations

[itex]g:R^2 \rightarrow R^2 [/itex] is the polar coordinate transformation defined by [itex] g(r, \theta ) = (r\cos \theta , r\sin \theta ). [/itex]

3. The attempt at a solution

My thought is to use the sequence of sets [itex] A_n = \{(r,\theta )| 1<r<n, 0<\theta < 2\pi \}[/itex] in the polar plane whose infinite union will equal [itex] R^2 - \overline{U} [/itex], where I can then show that [itex]\int _{A_N} f [/itex] is unbounded, implying [itex] \int _{R^2 - \overline{U}} f[/itex] does not exist. However, in order to use the polar coordinate transform, I need to show that [itex]g [/itex] is a diffeomorphism. However, I'm not sure how to show that g is bijective. If anyone has any advice, I would be very grateful. Thanks!

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# Homework Help: Existence of improper integral using polar transform (Munkres Analysis on Manifolds)

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