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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2015/12/d/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2015 / 12 / d / Solution by

Codex  0 2026-10-25

Both paths start at

(0, 0, 0)

and end at

(1, 2, 1)

. The potential values are

ϕ (0, 0, 0) = C

and

ϕ (1, 2, 1) = 4 + C

, so

Δ ϕ = 4 = \int_{C_{1}} F \cdot d r = \int_{C_{2}} F \cdot d r .

(1)

This is the fundamental theorem for line integrals: an exact one-form integrates to the difference of its single-valued potential at the endpoints. The agreement is guaranteed for every piecewise smooth connecting path, rather than being an accident of these two examples. In particular the integral around a closed path assembled from one route and the reverse of the other is zero.

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