 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013/1/solution

Work where

x \neq = 0

and

f

is differentiable. Set

q = y / x

. The chain rule gives

w_{x} = f^{'} (q) (- \frac{y}{x ^{2}}), w_{y} = f^{'} (q) \frac{1}{x} .

Consequently

x w_{x} + y w_{y} = - \frac{y}{x} f^{'} (q) + \frac{y}{x} f^{'} (q) = 0.

Geometrically, a radial rescaling of both coordinates leaves

y / x

unchanged, so the function has zero rate of change along radial dilations. No value or derivative at

x = 0

is implied by this representation.

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