Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 1 / iii / Solution

Use the full field in the PDF, including the exponential term missing from the transcription. Its divergence is

\nabla \cdot F = [z sin (yz) + 3 x^{2}] - z sin (yz) + 3 y^{2} = 3 (x^{2} + y^{2}) .

The exponential in the third component has zero

z

derivative. In cylindrical coordinates, the solid has

0 \leq r \leq 2

and

0 \leq z \leq 4 - r^{2}

. Therefore the divergence theorem gives the outward flux

\int_{S} F \cdot \overset{n}{^} d S = \int_{0}^{2 π} \int_{0}^{2} \int_{0}^{4 - r^{2}} 3 r^{2} r d z d r d ϕ = 6 π \int_{0}^{2} (4 r^{3} - r^{5}) d r = 32 π .

The integration includes the entire closed boundary, both cap and base.

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