A function \(f(r, \theta)\) is transformed into \(g(u, s)\) by means of the relations \(r \cos \theta = 1/u\), \(\tan \theta = s\).
Prove that
- [(i)] \(r \frac{\partial f}{\partial r} = -u \frac{\partial g}{\partial u}\)
- [(ii)] \(\frac{\partial f}{\partial \theta} = us \frac{\partial g}{\partial u} + (1 + s^2) \frac{\partial g}{\partial s}\)
If \(f\) satisfies the differential equation
\[\cos \theta \frac{\partial^2 f}{\partial r \partial \theta} + r \sin \theta \frac{\partial^2 f}{\partial r^2} = 0,\]
prove that
\[\frac{\partial g}{\partial u} = (1 + s^2)^\frac12 \phi(u),\]
where \(\phi\) is some function of \(u\).