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Calculate $\iint_S (\nabla \times F)\cdot dS= 0$ if $F= i+j+k$ where $\omega$ is the unit cube. First solve directly the integral and then use the Divergence Theorem.

In the firs part I do not know how to evaluate the integral directly I am confused about because in the Divergence Theorem we use $\iiint_V\nabla\cdot F$ instead of $\iint_S (\nabla \times F)$

Any hint or idea?

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  • $\begingroup$ Stokes Theorem... $\endgroup$
    – DiegoMath
    May 22 '18 at 18:25
  • $\begingroup$ But it says I Should use Divergence Theorem $\endgroup$ May 22 '18 at 18:28
  • $\begingroup$ Look the answer. $\endgroup$
    – DiegoMath
    May 22 '18 at 18:29
  • $\begingroup$ Remember that $\nabla\cdot(\nabla\times\vec{F})\equiv0$. $\endgroup$
    – DiegoMath
    May 22 '18 at 18:31
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If ${\bf F}={\bf i}+{\bf j}+{\bf k}$ then $\nabla\times{\bf F}={\bf 0}$, whatever $\nabla\times{\bf F}$ means. The $\omega$ (why didn't you use $\pi$, $e$, or $5$ to denote the unit cube?) does not occur in $\int_?\int_S (\nabla\times{\bf F})\cdot{\rm d}S$, but this does not play a rôle under the circumstances of this lousy question.

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Remember that $${\rm div}({\rm curl} \vec{F})\equiv0.$$

Using the Divergence theorem, the result follows.

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Since F= i+ j+ k is constant vector, $\nabla\times F$ is the 0 vector. The integral over any surface is 0.

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