A test density is defined on the unit square (metres) by (x,y) = cases 1

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 test density ρ(x,y)\rho(x,y) is defined on the unit square R=[0,1]×[0,1]R=[0,1]\times[0,1] (metres) by ρ(x,y)={1 kg m−2if both x and y are rational numbers0otherwise.\rho(x,y) = \begin{cases}1\,\mathrm{kg\,m^{-2}} & \text{if both $x$ and $y$ are rational numbers}\\ 0 & \text{otherwise.}\end{cases} This is a purely mathematical construction, not a claim about any real material. ((a)) Someone argues: "ρ\rho satisfies 0≤ρ≤1 kg m−20\le\rho\le1\,\mathrm{kg\,m^{-2}} everywhere on the bounded region RR, so the double integral ∬Rρ dA\iint_R\rho\,\mathrm{d}A (the total mass) exists." Explain why this is not enough, by describing the upper and lower sums of ρ\rho over an arbitrary partition of RR into boxes. ((b)) State whether ∬Rρ dA\iint_R\rho\,\mathrm{d}A exists as a (proper) Riemann double integral, and say whether the zero-content criterion could have decided this.

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