Flux through Unit-Sphere

Calculus Level pending

Find the flux of the vector field F ⃗ = ( F x , F y , F z ) = ( x 3 , y 3 , z 3 ) \vec{F} = (F_x,F_y,F_z) = (x^3,y^3,z^3) through a unit-sphere centered on the origin. If the answer can be expressed as A B π \frac{A}{B} \pi , where A A and B B are positive co-prime integers, determine A + B A+B .


The answer is 17.

Steven Chase by Steven Chase , 37, USA

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1 solution

Otto Bretscher
Dec 3, 2018

∫ S F ⃗ ⋅ d S ⃗ = ∫ S ( x 3 , y 3 , z 3 ) ⋅ ( x , y , z ) d S = ∫ S ( x 4 + y 4 + z 4 ) d S = 3 ∫ z 4 d S = 3 ∫ 0 2 π ∫ 0 π cos ⁡ 4 ϕ sin ⁡ ϕ d ϕ d θ = 3 × 2 π × 2 5 = 12 π 5 \int_S \vec{F} \cdot d\vec{S}=\int_S (x^3,y^3,z^3) \cdot (x,y,z) dS =\int_S (x^4+y^4+z^4)dS=3\int z^4 dS=3\int_{0}^{2\pi}\int_{0}^{\pi}\cos^4\phi \sin\phi d\phi d\theta=3\times 2\pi \times \frac{2}{5}=\frac{12\pi}{5}

The answer is 12 + 5 = 17 12+5=\boxed{17} . (Alternatively, we can use Ostrogradsky's theorem.)

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