Paper 3, Section II, B

Vector Calculus | Part IA, 2019

Define the Jacobian, JJ, of the one-to-one transformation

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

Give a careful explanation of the result

Df(x,y,z)dxdydz=ΔJϕ(u,v,w)dudvdw\int_{D} f(x, y, z) d x d y d z=\int_{\Delta}|J| \phi(u, v, w) d u d v d w


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))

and the region DD maps under the transformation to the region Δ\Delta.

Consider the region DD defined by

x2+y2k2+z21x^{2}+\frac{y^{2}}{k^{2}}+z^{2} \leqslant 1


x2α2+y2k2α2z2γ21\frac{x^{2}}{\alpha^{2}}+\frac{y^{2}}{k^{2} \alpha^{2}}-\frac{z^{2}}{\gamma^{2}} \geqslant 1

where α,γ\alpha, \gamma and kk are positive constants.

Let D~\tilde{D} be the intersection of DD with the plane y=0y=0. Write down the conditions for D~\tilde{D} to be non-empty. Sketch the geometry of D~\tilde{D} in x>0x>0, clearly specifying the curves that define its boundaries and points that correspond to minimum and maximum values of xx and of zz on the boundaries.

Use a suitable change of variables to evaluate the volume of the region DD, clearly explaining the steps in your calculation.

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