Exercise · ZW-STOKES-THEOREM-009 · ★★★ Advanced
If \(\nabla\times\mathbf F=\mathbf 0\) on all of \(\mathbb R^3\), which statement is correct?
Choose one option.
A \(\mathbf F=\mathbf 0\) B The flux of \(\mathbf F\) through every closed surface is \(0\) C The circulation of \(\mathbf F\) is \(0\) around every smooth closed curve D \(\mathbf F\) is constant Show answer C. The circulation of \(\mathbf F\) is \(0\) around every smooth closed curve← Back to the question
Explanation By Stokes, the circulation is the flux of \(\nabla\times\mathbf F=\mathbf 0\) through a surface bounded by \(C\). Zero flux of \(\mathbf F\) is a matter of divergence , not of curl .
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 ZW-STOKES-THEOREM-001 Let S be the disk x^2+y^2≤1 in the plane z=0, with normal mathbf k. Compute iint_S(∇×mathbf F)· dma…ZW-STOKES-THEOREM-002 Let mathbf F=langle z,x,yrangle and C the boundary of the triangle of the plane x+y+z=1 in the firs…ZW-STOKES-THEOREM-003 With the same curve C as in the previous exercise, compute oint_Cmathbf F· dmathbf r for mathbf F=l…ZW-STOKES-THEOREM-004 Let mathbf F=langle 2xyz,\;x^2z,\;x^2yrangle. What is oint_Cmathbf F· dmathbf r over any smooth clo…ZW-STOKES-THEOREM-005 If S lies in the xy-plane with normal mathbf k and mathbf F=langle P,Q,0rangle, what does (∇×mathbf…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…