Exercise · ZW-DIVERGENCE-THEOREM-004 · ★☆☆ Basic
Compute the outward flux of \(\mathbf F=\langle x,y,z\rangle\) through the sphere of radius \(2\).
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- D. \(32\pi\)
Explanation
\(\nabla\cdot\mathbf F=3\) and the ball of radius \(2\) has volume \(\tfrac{4\pi}{3}\cdot8=\tfrac{32\pi}{3}\); the flux is \(3\cdot\tfrac{32\pi}{3}=32\pi\).
Review the concept: Divergence Theorem
The divergence theorem relates the flux of a field \(\mathbf F\) through a closed surface \(S\), with outward normal, to the triple integral of its divergence over the solid \(E\) that \(S\) encloses.
\[\iint_S\mathbf F\cdot d\mathbf S=\iiint_E\nabla\cdot\mathbf F\,dV\]
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