 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2021/10/ii/solution

Assume

a > 0

and choose the outward orientation. On the sphere,

n = (x, y, z) / a

, so

F \cdot n = (x^{2} + y^{2} + 2 z^{2}) / a = (a^{2} + z^{2}) / a

z = a cos θ

and

d S = a^{2} sin θ d θ d ϕ

\iint_{S} F \cdot n d S = a^{3} \int_{0}^{2 π} d ϕ \int_{0}^{π} (1 + cos^{2} θ) sin θ d θ = 2 π a^{3} (2 + \frac{2}{3}) = \frac{16 π a ^{3}}{3} .

Independently,

\nabla \cdot F = 1 + 1 + 2 = 4

. Its volume integral over the ball is

∭_{V} 4 d V = 4 \int_{0}^{a} r^{2} d r \int_{0}^{π} sin θ d θ \int_{0}^{2 π} d ϕ = \frac{16 π a ^{3}}{3} .

The two calculations agree exactly with the divergence theorem. The expression requested is a scalar flux of a vector field, namely the dot product with the directed surface area element.

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