Consider the function on the square

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Calculus of change and accumulation

Conditions of Validity, Ch Multiple integrals: Fubini, the change-of-variables theorem and integrals that do not exist

Problem

Consider the function g(x,y)=x2−y2(x2+y2)2g(x,y) = \dfrac{x^2-y^2}{(x^2+y^2)^2} on the square [0,1]×[0,1][0,1]\times[0,1], with g(0,0)g(0,0) left undefined (this is a purely mathematical construction, so it is allowed to be dimensionless). ((a)) Compute ∫01(∫01g(x,y) dy)dx\displaystyle\int_0^1\left(\int_0^1 g(x,y)\,\mathrm{d}y\right)\mathrm{d}x. ((b)) Compute ∫01(∫01g(x,y) dx)dy\displaystyle\int_0^1\left(\int_0^1 g(x,y)\,\mathrm{d}x\right)\mathrm{d}y. ((c)) Someone argues: "both iterated integrals were computed correctly and each is a perfectly good number, so the double integral ∬[0,1]2g dA\iint_{[0,1]^2}g\,\mathrm{d}A exists and equals one of them, whichever order is more natural." Find the error, and name the hypothesis of Fubini's theorem that fails here.

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