Exercise · ZW-DIVERGENCE-THEOREM-001 · ★☆☆ Basic
Compute the outward flux of \(\mathbf F=\langle x,y,z\rangle\) through the unit sphere \(x^2+y^2+z^2=1\).
Choose one option.
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- C. \(4\pi\)
Explanation
\(\nabla\cdot\mathbf F=3\), so the flux is \(3\) times the volume of the ball, \(3\cdot\tfrac{4\pi}{3}=4\pi\). The typical mistake is forgetting the factor \(3\) and answering \(\tfrac{4\pi}{3}\).
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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