Exercise · ZW-STOKES-THEOREM-005 · ★☆☆ Basic
If \(S\) lies in the \(xy\)-plane with normal \(\mathbf k\) and \(\mathbf F=\langle P,Q,0\rangle\), what does \((\nabla\times\mathbf F)\cdot\mathbf k\) reduce to?
Choose one option.
A \(P_x+Q_y\) B \(P_y-Q_x\) C \(Q_x-P_y\) D \(P+Q\) Show answer ← Back to the question
Explanation The \(\mathbf k\)-component of the curl is \(Q_x-P_y\); that is why Green's theorem is the planar case of Stokes.
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 \(S\).
\[\oint_C\mathbf F\cdot d\mathbf r=\iint_S(\nabla\times\mathbf F)\cdot d\mathbf S\]
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