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I've found the following proof as an exercise left to the reader in Griffith's "Introduction to Electronidamics", and I would like to know whether what I've done is right or not (That's the first time I work with gradients since I am an high-schooler). $\int_{Volume}\nabla \cdot \vec{v} d\tau=\oint_{Surface} \vec{v} \cdot \vec{ds}$ making the substitution $\vec{v}=\vec{c}T$ where c is a constant vector and $T=T(\vec{r})$, it follows from the divergence theorem that $\int_{V}\nabla \cdot \vec{v}d\tau=\int_{V} \nabla \cdot (\vec{c}T)d\tau=\oint T \vec{c} \cdot \vec{da}$ from which, since c is constant, $\vec{c} \cdot \int_{V} \nabla Td\tau=\vec{c} \cdot \oint T\vec{da}$, from which it follows the thesis

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  • $\begingroup$ P.S is it always possible to take a constant vector out of the integral sign, thus doing the integral first and the dot product with the constant vector after? $\endgroup$ Commented Mar 11, 2021 at 19:37

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