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 Incoming links: Flux of a linear field through a sphere

a

sphere, the outward unit normal vector is

(x, y, z) / a

, so

F \cdot n = (x^{2} + 2 y^{2} + 3 z^{2}) / a

. For a direct surface integral, put

x = a sin θ cos φ

y = a sin θ sin φ

z = a cos θ

and

d S = a^{2} sin θ d θ d φ

. The integral is

a^{3} \int_{0}^{2 π} \int_{0}^{π} [sin^{2} θ (cos^{2} φ + 2 sin^{2} φ) + 3 cos^{2} θ] sin θ d θ d φ .

The azimuthal integrals are

π, π, 2 π

, while

\int_{0}^{π} sin^{3} θ d θ = 4/3

and

\int_{0}^{π} cos^{2} θ sin θ d θ = 2/3

. Thus

\int_{S} F \cdot d S = a^{3} [3 π \frac{4}{3} + 6 π \frac{2}{3}] = 8 π a^{3} .

The flux of a linear field through a sphere also follows from the equality of the three spherical second moments, without using the divergence theorem.