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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/20/b/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 20 / b / Solution by

Codex  0 2026-10-18

For a sufficiently smooth vector field on a volume

V

and its closed boundary

S

, the divergence theorem states

\int_{S} v \cdot n d S = \int_{V} \nabla \cdot v d V,

(1)

with the unit normal directed outwards. Here

\nabla \cdot v = \partial_{x} (- α yz) + \partial_{y} (αx z) + \partial_{z} 0 = 0.

(2)

Hence the net volume flux through every closed surface in the liquid is zero. This verifies incompressible flow directly. It does not require the velocity to vanish pointwise on an arbitrary closed surface; inward and outward contributions can cancel.

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