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I have seen a proof that concludes this:

$\iiint_{V} \nabla \times \mathbf{B} \, dV = \iint_{S} \mathbf{n} \times \mathbf{B}\,dS$

My question is: if is it possible to take the volume integral of a vector field? Because $\nabla \times \mathbf{B}$ is

link of the book (pg 121): http://www.uop.edu.pk/ocontents/Vector%20Analysis%20Schaum.pdf

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    $\begingroup$ If one integrates a vector valued function, this is usually defined to be component wise, i.e. you would do the volume integral for each component of $\nabla \times \mathbf{B}$. $\endgroup$
    – GG314
    Commented Apr 9 at 5:53

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