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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/ia/paper-3/10a/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 3 / 10A / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-06

For an oriented piecewise smooth surface

S

and a continuously differentiable vector field on a neighborhood of it, Stokes theorem states

\int_{S} (\nabla \times F) \cdot d S = \oint_{\partial S} F \cdot d x .

(1)

The oriented surface element and boundary direction must agree by the right-hand rule. Every boundary component is included; for an upward-oriented graph over an annulus the outer boundary is anticlockwise and the inner boundary clockwise when viewed from above.

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