Exercise · ZW-STOKES-THEOREM-003 · ★☆☆ Basic
With the same curve \(C\) as in the previous exercise, compute \(\oint_C\mathbf F\cdot d\mathbf r\) for \(\mathbf F=\langle y,z,x\rangle\).
Choose one option.
Show answer
- D. \(-\tfrac32\)
Explanation
\(\nabla\times\mathbf F=\langle-1,-1,-1\rangle\); its product with \(\langle1,1,1\rangle\) is \(-3\) and the projected area is \(\tfrac12\), so the result is \(-\tfrac32\).
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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- ZW-STOKES-THEOREM-006 Let S be the hemisphere x^2+y^2+z^2=1, z≥0, with upward normal, and mathbf F=langle y,-x,0rangle. C…
- ZW-STOKES-THEOREM-007 If iint_S(∇×mathbf F)· dmathbf S=4 with S oriented upward, what is it after reversing the orientati…
- ZW-STOKES-THEOREM-008 Let C be the circle x^2+y^2=4 in the plane z=3, counterclockwise seen from above, and mathbf F=lang…
- ZW-STOKES-THEOREM-009 If ∇×mathbf F=mathbf 0 on all of mathbb R^3, which statement is correct?