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In class, we were asked to prove the identity $\nabla\times(\Omega\vec{V})=\Omega(\nabla\times\vec{V})-\vec{V}\times\nabla\Omega$. One of the possible approaches involved separating the del operator into two operators,

\begin{equation}\nabla\equiv\nabla_{\Omega}+\nabla_{\vec{V}}\end{equation}

where $\nabla_{\Omega}$ differentiates $\Omega$ and leaves it operates on $\vec{V}$ zero, and vice versa. I know how to prove the identity, once I accept that $\nabla\equiv\nabla_{\Omega}+\nabla_{\vec{V}}$, but I find ti difficult to convince myself that it is true. Can del be separated like this?

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  • $\begingroup$ Please let me know how to improve my answer. I really want to give you the best answer you can. And feel free to up vote and accept an answer as you see fit. $\endgroup$
    – Mark Viola
    Apr 2, 2021 at 21:36

1 Answer 1

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The Del operator $\nabla$ operates on the independent coordinate variables, $x$, $y$, and $z$,not the dependent variables $\vec V(x,y,z)$ and $\Omega(x,y,z)$. So, we have

$$\begin{align} \nabla \times (\Omega \vec V)&=\sum_{i=1}^3 \hat x_i \frac{\partial}{\partial x_i}\times \left(\Omega \sum_{j=1}^3 \hat x_jV_j\right)\\\\ &=\sum_{i=1}^3 \sum_{j=1}^3 (\hat x_i \times \hat x_j )\left(\Omega \frac{\partial V_j}{\partial x_i}+V_j \frac{\partial \Omega}{\partial x_i}\right)\\\\ &=\Omega \nabla \times \vec V+\nabla \Omega \times \vec V\\\\ &=\Omega \nabla \times \vec V-\vec V\times \nabla \Omega \end{align}$$

as was to be shown!

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  • $\begingroup$ This doesn't really answer the question. $\endgroup$
    – TonyK
    Mar 7, 2021 at 17:54
  • $\begingroup$ @TonyK It certainly does. The question is "Can the del operator be “separated” like this?" The solution posted herein answers that with a "No." Read "not the dependent variables." $\endgroup$
    – Mark Viola
    Mar 7, 2021 at 19:06
  • $\begingroup$ @NX37B Please let me know how to improve my answer. I really want to give you the best answer you can. And feel free to up vote and accept an answer as you see fit. $\endgroup$
    – Mark Viola
    Jun 2, 2021 at 16:29

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