Flux through Unit-Sphere (Part 2)

Calculus Level pending

Find the flux of the vector field F ⃗ = ( F x , F y , F z ) = ( x y z , x + y + z , 0 ) \vec{F} = (F_x,F_y,F_z) = (x y z,x+y+z, 0) 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 7.

Steven Chase by Steven Chase , 37, USA

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1 solution

Otto Bretscher
Dec 3, 2018

By Ostrogradsky's Theorem,

∫ S = ∂ B F ⃗ ⋅ d S ⃗ = ∫ B ∇ ⃗ ⋅ F ⃗ d V = ∫ B ( y z + 1 ) d V = ∫ B d V = 4 π 3 \int_{S=\partial{B}}\vec{F} \cdot d\vec{S}=\int_B \vec{\nabla} \cdot \vec{F}\hspace{1mm} dV=\int_B(yz+1)dV=\int_B dV=\frac{4\pi}{3}

The answer is 4 + 3 = 7 4+3=\boxed{7} . Note that ∫ B y z d V = 0 \int_B yz\hspace{1mm} dV =0 by symmetry.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice solution! Read Stewart Calculus. It provides a solution from first principles.

Krishna Karthik - 2 years, 6 months ago

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