Paper 3, Section II, C

Vector Calculus | Part IA, 2013

Give an explicit formula for J\mathcal{J} which makes the following result hold:

Dfdxdydz=DϕJdudvdw\int_{D} f d x d y d z=\int_{D^{\prime}} \phi|\mathcal{J}| d u d v d w

where the region DD, with coordinates x,y,zx, y, z, and the region DD^{\prime}, with coordinates u,v,wu, v, w, are in one-to-one correspondence, and

ϕ(u,v,w)=f(x(u,v,w),y(u,v,w),z(u,v,w))\phi(u, v, w)=f(x(u, v, w), y(u, v, w), z(u, v, w))

Explain, in outline, why this result holds.

Let DD be the region in R3\mathbb{R}^{3} defined by 4x2+y2+z294 \leqslant x^{2}+y^{2}+z^{2} \leqslant 9 and z0z \geqslant 0. Sketch the region and employ a suitable transformation to evaluate the integral

D(x2+y2)dxdydz\int_{D}\left(x^{2}+y^{2}\right) d x d y d z

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