Past exam of the mathematical methods II course of the University of Oxford / 2017 / 2

Set $x=au$, $y=bv$, $z=cw$. For nonzero $a,b,c$, the absolute Jacobian determinant is $|abc|$, and the transformed interior is the ball $u^2+v^2+w^2\le 4$. Its radius is two, soIf $a,b,c$ are taken as positive semiaxis parameters, this is $32\pi abc/3$. The actual semiaxes are $2|a|,2|b|,2|c|$; omitting the factor two in the transformed radius would make the volume eight times too small.