 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/11/b/i/solution

r^{2} = x^{2} + y^{2} + z^{2}

, the gradient is

v = 2 e^{r^{2}} (x, y, z)

. Its divergence, equivalently the Laplacian of

ϕ

, is

\nabla \cdot v = (6 + 4 r^{2}) e^{r^{2}} .

On the unit sphere this scalar is the constant

10 e

2 π

, so

\int_{S} (\nabla \cdot v) d S = 20 π e .

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