Exercise · ZW-STOKES-THEOREM-004 · ★☆☆ Basic
Let \(\mathbf F=\langle 2xyz,\;x^2z,\;x^2y\rangle\). What is \(\oint_C\mathbf F\cdot d\mathbf r\) over any smooth closed curve \(C\) in space?
Choose one option.
Show answer
- B. \(0\)
Explanation
\(\mathbf F=\nabla(x^2yz)\), so \(\nabla\times\mathbf F=\mathbf 0\) and, by Stokes, the circulation around any closed curve is \(0\).
Review the concept: Stokes' Theorem
Stokes' theorem relates the circulation of a field \(\mathbf F\) along the closed curve \(C\) bounding an oriented surface \(S\) to the flux of the curl of \(\mathbf F\) through \(S\).
\[\oint_C\mathbf F\cdot d\mathbf r=\iint_S(\nabla\times\mathbf F)\cdot d\mathbf S\]
More practice problems: stokes' theorem
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