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 Incoming links: Volume enclosed by a sphere

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 6 / c / Solution Created 2026-10-18 Updated 2026-10-20

The divergence is

\nabla \cdot F = 1 + 2 + 3 = 6

. Since the volume enclosed by a sphere is

4 π a^{3} /3

,

\int_{V} \nabla \cdot F d V = 6 \frac{4 π a ^{3}}{3} = 8 π a^{3} .

This agrees with the independently computed outward surface integral, as the divergence theorem requires.

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