Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 1 / 6 / a

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- Solution a

Solution

Take the outward orientation of a surface. The divergence is

\nabla \cdot F = y^{2} + z^{2} + x^{2}

, so the divergence theorem converts the closed surface flux into an integral over the box:

\int_{S} F \cdot d S = \int_{- a}^{a} \int_{- b}^{b} \int_{- c}^{c} (x^{2} + y^{2} + z^{2}) d z d y d x .

Each term separates; for example the

x^{2}

term is

(2 a^{3} /3) (2 b) (2 c)

. Summing gives

\int_{S} F \cdot d S = \frac{8 ab c}{3} (a^{2} + b^{2} + c^{2}) .

Here

a, b, c > 0

are the box half-lengths. Reversing orientation of a surface would negate the result.

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