Exercise · ZW-DIVERGENCE-THEOREM-002 · ★☆☆ Basic
Let \(\mathbf F=\langle x^2,y^2,z^2\rangle\) and \(S\) the boundary of the cube \([0,1]^3\) with outward normal. Compute \(\iint_S\mathbf F\cdot d\mathbf S\).
Choose one option.
Show answer
- A. \(3\)
Explanation
\(\nabla\cdot\mathbf F=2(x+y+z)\) and \(\iiint_{[0,1]^3}2(x+y+z)\,dV=2\cdot\tfrac32=3\). Computing the six faces separately gives the same value with more work.
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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