Exercise · ZW-DIVERGENCE-THEOREM-006 · ★★☆ Intermediate
Compute the outward flux of \(\mathbf F=\langle x^3,y^3,z^3\rangle\) through the unit sphere.
Choose one option.
Show answer
- B. \(\tfrac{12\pi}{5}\)
Explanation
\(\nabla\cdot\mathbf F=3(x^2+y^2+z^2)=3\rho^2\) in spherical coordinates, and \(\int_0^{2\pi}\!\int_0^{\pi}\!\int_0^1 3\rho^2\cdot\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta=3\cdot\tfrac15\cdot2\cdot2\pi=\tfrac{12\pi}{5}\).
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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