Let be dimensionless real coordinates (this is a purely mathematical exercise

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

Let u,vu,v be dimensionless real coordinates (this is a purely mathematical exercise, so no physical unit is attached) and let Φ:R2→R2\Phi:\mathbb{R}^2\to\mathbb{R}^2 be a map that is continuously differentiable near a point (u0,v0)(u_0,v_0), written Φ(u,v)=(x(u,v),y(u,v))\Phi(u,v)=(x(u,v),y(u,v)). ((a)) Starting from the linear approximation Φ(u0+du, v0+dv)=Φ(u0,v0)+DΦ(u0,v0)(dudv)+o(∥(du,dv)∥)\Phi(u_0+\mathrm{d}u,\,v_0+\mathrm{d}v)=\Phi(u_0,v_0)+\mathrm{D}\Phi(u_0,v_0)\binom{\mathrm{d}u}{\mathrm{d}v}+o(\|(\mathrm{d}u,\mathrm{d}v)\|), show that the small rectangular cell of sides du,dv>0\mathrm{d}u,\mathrm{d}v>0 at (u0,v0)(u_0,v_0) has image area equal to ∣J(u0,v0)∣ du dv|J(u_0,v_0)|\,\mathrm{d}u\,\mathrm{d}v to leading order, where J=det⁡DΦ=∂(x,y)/∂(u,v)J=\det\mathrm{D}\Phi=\partial(x,y)/\partial(u,v), and say what happens to the neglected terms as du,dv→0\mathrm{d}u,\mathrm{d}v\to0. ((b)) Apply this to the map Φ(u,v)=(u2−v2, 2uv)\Phi(u,v)=(u^2-v^2,\,2uv): find J(u,v)J(u,v) explicitly. ((c)) Use the result of part (b) to compute the area of the image of the quarter-disc region D={(u,v):u≥0, v≥0, u2+v2≤4}D=\{(u,v): u\ge0,\ v\ge0,\ u^2+v^2\le 4\} under Φ\Phi. ((d)) Check the answer to part (c) by identifying the image of DD geometrically, using the substitution u=rcos⁡θu=r\cos\theta, v=rsin⁡θv=r\sin\theta, and computing its area directly.

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