Consider the map with ,

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

Consider the map Φ(u,v)=(x,y)\mathbf{\Phi}(u,v) = (x,y) with x=u2−v2x = u^2 - v^2, y=2uvy = 2uv, applied to the quarter-disc D={(u,v):u>0, v>0, u2+v2≤1}.D = \{(u,v) : u > 0,\ v > 0,\ u^2+v^2 \le 1\}. ((a)) Compute the Jacobian matrix DΦ\mathrm{D}\mathbf{\Phi} and its determinant, and use the inverse function theorem to say at which points of the plane Φ\mathbf{\Phi} is locally invertible. ((b)) Writing u=rcos⁡θu = r\cos\theta, v=rsin⁡θv=r\sin\theta, show that Φ\mathbf{\Phi} maps DD, with the origin removed, bijectively onto the upper half of the unit disc H={(x,y):x2+y2≤1, y≥0}H = \{(x,y): x^2+y^2 \le 1,\ y \ge 0\} with the origin removed (a shared boundary ray may be disregarded as it has zero content). ((c)) Use Φ\mathbf{\Phi} to evaluate I=∬Hdx dyx2+y2I = \iint_H \frac{\mathrm{d}x\,\mathrm{d}y}{\sqrt{x^2+y^2}} by excising a small disc of radius ϵ\epsilon about the origin from DD, applying the change-of-variables theorem on the remaining region, and letting ϵ→0\epsilon \to 0.

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