Exercise · ZW-STOKES-THEOREM-010 · ★★★ Advanced
Let \(\mathbf F=\langle 0,0,xy\rangle\) and \(C\) the unit circle in the \(xy\)-plane, counterclockwise. What is \(\oint_C\mathbf F\cdot d\mathbf r\)?
Choose one option.
Show answer
- D. \(0\)
Explanation
Here \(\nabla\times\mathbf F=\langle x,-y,0\rangle\) and, over the disk with normal \(\mathbf k\), its product with \(\mathbf k\) is \(0\). Directly: \(d\mathbf r\) has no \(z\)-component, so \(\mathbf F\cdot d\mathbf r=0\).
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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