Vector Calculus | Part IA, 2003

Write down an expression for the Jacobian JJ of a transformation

(x,y,z)(u,v,w)(x, y, z) \rightarrow(u, v, w)

Use it to show that

Dfdxdydz=ΔϕJdudvdw\int_{D} f d x d y d z=\int_{\Delta} \phi|J| d u d v d w

where DD is mapped one-to-one onto Δ\Delta, 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))

Find a transformation that maps the ellipsoid DD,

x2a2+y2b2+z2c21\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} \leqslant 1

onto a sphere. Hence evaluate

Dx2dxdydz\int_{D} x^{2} d x d y d z

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