An interesting problem for Multivariable Calculus Students

This is aimed a bit more at students, rather than helpers, but I won't go so far as to ask helpers to refrain from answering.

The divergence theorem states that

$\displaystyle \iiint_{V}\nabla\cdot \mathbf{F}\,dV=\iint_{{\partial}V}\mathbf{F}\cdot d\mathbf{S},$

and the Kelvin-Stokes theorem states that

$\displaystyle \iint_{A}\nabla\times\mathbf{G}\cdot d\mathbf{S}=\oint_{\partial A}\mathbf{G}\cdot d\mathbf{r}.$

If I set

$\displaystyle \mathbf{F}=\nabla\times\mathbf{G},$

then comment on the legitimacy of writing

$\displaystyle \iiint_{V}\nabla\cdot(\nabla\times\mathbf{G})\,dV= \iint_{\partial V}\nabla\times\mathbf{G}\cdot d\mathbf{S}=\oint_{\partial(\partial V)}\mathbf{G}\cdot d\mathbf{r}.$

Re: An interesting problem for Multivariable Calculus Students

I asked the exact same question in Calculus Forum.

personally i do not know the correct answer. But div(curl(G)) is always zero, making the far right side always zero

whereas the far left side is only zero if G field is conservative. so I guess I need better brain to prove this.....

I do not know the answer.

Re: An interesting problem for Multivariable Calculus Students

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