 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/16/b/iii/solution

Add the four outward fluxes:

Φ_{closed} = - \frac{1}{6} - \frac{1}{4} - \frac{1}{3} + 1 = \frac{1}{4} .

For a check,

\nabla \cdot F = 2 (x + y + z)

. The divergence theorem gives the same result from

∭_{x, y, z \geq 0, x + y + z \leq 1} 2 (x + y + z) d x d y d z = \frac{1}{4} .

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