 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/12/b/ii/solution

On a chart with

x \neq = 0

f (x, y) = arctan (\frac{b y}{a x}) + C .

The chain rule verifies both components:

f_{x} = \frac{1}{1 + ( b y / ( a x ) ) ^{2}} (- \frac{b y}{a x ^{2}}) = - \frac{ab y}{D} = P, f_{y} = \frac{1}{1 + ( b y / ( a x ) ) ^{2}} \frac{b}{a x} = \frac{ab x}{D} = Q .

A coordinate-independent description of the local potential is a continuous branch of

ar g (a x + ib y)

, the complex argument of the scaled coordinates. Such branches can also cover points where

x = 0

, using a different angular chart. Continuing once around the missing origin changes that angle by a nonzero multiple of

2 π

, so there is no global single-valued $f$ on the full stated domain.

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