Discussion
A set of all surfaces of revolution with \(z\)-axis as the axis of revolution is characterized by the partial differential equation :
Answer: C
The equation is \(y\,\partial z/\partial x - x\,\partial z/\partial y = 0\).
A surface of revolution about the \(z\)-axis has the form \(z = f(x^{2} + y^{2})\). Differentiating,
\[ \begin{aligned} z_{x} &= 2x f'(x^{2} + y^{2}) \\ z_{y} &= 2y f'(x^{2} + y^{2}) \end{aligned} \]
Eliminating \(f'\): \(y z_{x} = 2xy f' = x z_{y}\), so \(y z_{x} - x z_{y} = 0\).
The other combinations do not cancel: \(x z_{x} + y z_{y} = 2(x^{2} + y^{2})f'\), and \(x z_{x} - y z_{y}\) and \(y z_{x} + x z_{y}\) leave terms in \(4xyf'\) or \(2(x^{2} - y^{2})f'\).
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