Exercise 2224Calculus › Multivariable calculus › Vector calculusM1Added 3 Nov 2026

Stokes on a torus (statement)

#Stokes#torus#closed surface

Statement

Show that the flux of any curl field through a closed surface is zero.

Hint

$\iint_S(\nabla\times\vec F)\cdot\vec n\,dS=\iiint_V\operatorname{div}(\nabla\times\vec F)\,dV=0$ .

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Divergence of curl. $\operatorname{div}(\nabla\times\vec F)=0$ .

Apply divergence theorem. $\iint_S(\nabla\times\vec F)\cdot\vec n\,dS=\iiint_V0\,dV=0$ .

Conclusion. Any curl field has zero flux through any closed surface.