Exercise · ZW-DIVERGENCE-THEOREM-003 · ★☆☆ Basic
Compute the outward flux of \(\mathbf F=\langle 2x,-y,3z\rangle\) through the boundary of the box \([0,1]\times[0,2]\times[0,3]\).
Choose one option.
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- B. \(24\)
Explanation
\(\nabla\cdot\mathbf F=2-1+3=4\), constant, and the box has volume \(6\): the flux is \(4\cdot6=24\).
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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