Someone evaluates

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

Someone evaluates I=∬S(x+y) dx dyI=\displaystyle\iint_{S} (x+y)\,\mathrm{d}x\,\mathrm{d}y, where S=[0,1]×[0,1]S=[0,1]\times[0,1] is the unit square, by the substitution x=vx=v, y=uy=u (swapping the two coordinates, an idealised, everywhere continuously differentiable, bijective map of SS onto itself). Their working reads: "∂(x,y)∂(u,v)=det⁡(0110)=−1\dfrac{\partial(x,y)}{\partial(u,v)} = \det\begin{pmatrix}0&1\\1&0\end{pmatrix} = -1, so dx dy=−du dv\mathrm{d}x\,\mathrm{d}y = -\mathrm{d}u\,\mathrm{d}v. Then I=∫01 ⁣ ⁣∫01(u+v)(−1) du dv=−(12+12)=−1.I = \int_0^1\!\!\int_0^1 (u+v)(-1)\,\mathrm{d}u\,\mathrm{d}v = -\left(\tfrac12+\tfrac12\right) = -1." ((a)) Find the error in this working. ((b)) State the correct value of II.

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