 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-1/15d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

B = \nabla \times A

, Stokes theorem gives, for any oriented surface

S

bounded by

C

Φ = \int_{S} B \cdot d S = \int_{S} (\nabla \times A) \cdot d S = \oint_{C} A \cdot d x .

A \mapsto A + \nabla χ

, the integral changes by

\oint_{C} \nabla χ \cdot d x = 0

because

C

is closed. The flux expression is therefore gauge independent.

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