A cylindrical hole of radius \(r\) is bored through a solid sphere of radius \(a\), the axis of the hole being along a diameter of the sphere. Find the volume and total surface area of the remaining portion of the sphere, and show that, for fixed \(a\), its surface area is maximum when \(r=a/2\).
The function \(u \equiv f(x_1, x_2, \dots, x_n)\) satisfies the identity \[ f(kx_1, k^2 x_2, \dots, k^n x_n) = k^a f(x_1, x_2, \dots, x_n) \] for fixed \(a\), all \(x_1, x_2, \dots, x_n\), and all positive values of \(k\). Show that \[ x_1 \frac{\partial u}{\partial x_1} + 2x_2 \frac{\partial u}{\partial x_2} + \dots + nx_n \frac{\partial u}{\partial x_n} = au. \] If, further, \(u\) is defined as a function of \(\xi\) and \(\eta\) by the substitutions \[ x_r = \xi^r + \eta^r \quad (r=1, 2, \dots, n), \] show that \[ \xi\frac{\partial u}{\partial \xi} + 2\eta\frac{\partial u}{\partial \eta} = -au. \]