A pattern function is defined on

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

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

Problem

A pattern function h(x,y)=xy(x2−y2)(x2+y2)3h(x,y) = \dfrac{xy(x^2-y^2)}{(x^2+y^2)^3} is defined on [0,1]×[0,1][0,1]\times[0,1], with h(0,0)h(0,0) left undefined (this is a purely mathematical construction, so it is allowed to be dimensionless). ((a)) Show that ∫01h(x,y) dy=x2(x2+1)2\displaystyle\int_0^1 h(x,y)\,\mathrm{d}y = \frac{x}{2(x^2+1)^2} for each x∈(0,1]x\in(0,1], and hence compute ∫01(∫01h(x,y) dy)dx\displaystyle\int_0^1\left(\int_0^1 h(x,y)\,\mathrm{d}y\right)\mathrm{d}x. ((b)) Using the symmetry h(y,x)=−h(x,y)h(y,x) = -h(x,y), state ∫01(∫01h(x,y) dx)dy\displaystyle\int_0^1\left(\int_0^1 h(x,y)\,\mathrm{d}x\right)\mathrm{d}y without repeating the calculation. ((c)) A claimed proof of the change-of-order rule for iterated integrals says: "both orders give finite numbers, therefore reversing the order is valid here." Explain what is wrong, referring to the size of hh near the origin.

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