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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014/12/c/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2014 / 12 / c / Solution by

Codex  0 2026-10-25

Integrate the first component of

F = \nabla ϕ

with respect to

x

:

ϕ = 3 x y^{2} - x^{3} yz - x z^{2} + g (y, z) .

Matching the

y

component gives

g_{y} = 0

, and matching the

z

component gives

g_{z} = 0

. Hence

ϕ = 3 x y^{2} - x^{3} yz - x z^{2} + C .

The fundamental theorem for line integrals now yields

I = ϕ (1, 1, 1) - ϕ (0, 0, 0) = 3 - 1 - 1 = 1.

The potential is a single-valued polynomial everywhere, so any piecewise smooth path with these endpoints has the same integral. The plus-gradient convention here should not be confused with the minus gradient of a mechanical potential energy.

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