91Ó°ÊÓ

Verify that for Cartesian coordinates the scale factors are all equal to 1 .

A coordinate system \((u, v, w)\) is related to Cartesian coordinates \(\left(x_{1}, x_{2}, x_{3}\right)\) by $$ x_{1}=u v w, \quad x_{2}=u v\left(1-w^{2}\right)^{1 / 2}, \quad x_{3}=\left(u^{2}-v^{2}\right) / 2 $$ (a) Find the scale factors \(h_{u}, h_{v}, h_{w}\). (b) Confirm that the \((u, v, w)\) system is orthogonal. (c) Find the volume element in the \((u, v, w)\) system.

A cylindrical apple corer of radius \(a\) cuts through a spherical apple of radius \(b\). How much of the apple does it remove?