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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013/9/b/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2013 / 9 / b / Solution by

Codex  0 2026-10-25

On the punctured plane

x^{2} + y^{2} > 0

, write

M = - y / (x^{2} + y^{2})

,

N = x / (x^{2} + y^{2})

. Direct differentiation gives

N_{x} = \frac{y ^{2} - x ^{2}}{( x ^{2} + y ^{2} ) ^{2}} = M_{y} .

There is no

z

dependence or

z

component, so

\nabla \times F = (0, 0, N_{x} - M_{y}) = 0 (x, y) \neq = (0, 0) .

The origin is excluded: the field is undefined there. A zero classical curl away from the origin does not state that the field is smooth on an entire disk containing it.

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