Exercise · ZW-STOKES-THEOREM-008 · ★★☆ Intermediate

Let \(C\) be the circle \(x^2+y^2=4\) in the plane \(z=3\), counterclockwise seen from above, and \(\mathbf F=\langle -y,x,z\rangle\). Compute \(\oint_C\mathbf F\cdot d\mathbf r\).

Choose one option.

Show answer
  • B. \(8\pi\)

Explanation

\(\nabla\times\mathbf F=\langle0,0,2\rangle\); over the disk \(x^2+y^2\le4\) at \(z=3\) the flux is \(2\cdot4\pi=8\pi\). The value of \(z\) changes nothing.

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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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