Exercise · ZW-DIVERGENCE-THEOREM-007 · ★★☆ Intermediate
Let \(\mathbf F=\langle xz,yz,z^2\rangle\) and \(S\) the closed surface of the cylinder \(x^2+y^2\le1\), \(0\le z\le2\) (with lids), outward normal. Compute the flux.
Choose one option.
A \(4\pi\) B \(2\pi\) C \(16\pi\) D \(8\pi\) Show answer ← Back to the question
Explanation \(\nabla\cdot\mathbf F=z+z+2z=4z\). In cylindrical coordinates : \(\int_0^2\!\int_0^{2\pi}\!\int_0^1 4z\,\rho\,d\rho\,d\theta\,dz=4\cdot\tfrac12\cdot2\pi\cdot2=8\pi\).
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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