$Problem:$ If a vector function $V=V(x,y,z)$ is not irrotational, show that if there exists a scalar function $g=g(x,y,z)$ such that $gV$ is irrotational, then $$V\cdot (\nabla \times V )=0$$

Remember, $V$ Irrotational $\iff \nabla\times V=0$

My attempt:

Since $\nabla\times V\not=0$, $\nabla\times V$ is a vector that is orthogonal to $V$, and so $V\cdot \nabla\times V=0$ by definition of the dot product.

I am totally unconvinced this can be the answer as nowhere did I use the fact that there exists a scalar function $g$ in my solution.

Any help is greatly appreciated!

  • $\begingroup$ V isn’t irrotational. gV is. $\endgroup$ – Jake Rose Oct 20 '19 at 15:12
  • $\begingroup$ $V$ not irrotational means $V$ is rotational... did I miss something? @JakeRose $\endgroup$ – Kam Oct 20 '19 at 15:15
  • $\begingroup$ @JakeRose So what does that tell me? $\endgroup$ – Kam Oct 20 '19 at 15:25
  • $\begingroup$ Might Mathematics be better suited for this math question? $\endgroup$ – Kyle Kanos Oct 20 '19 at 15:38
  • $\begingroup$ Related: physics.stackexchange.com/q/151030/2451 $\endgroup$ – Qmechanic Oct 20 '19 at 15:43

Since $g\vec{V}$ is irrotational, $\nabla\times(g\vec{V}) = 0$. Therefore, $\nabla g \times \vec{V} + g\nabla\times\vec{V} = 0$ or $\vec{V} \times \nabla g = g\nabla\times\vec{V}$. Now take the dot product with $\vec{V}$ so that \begin{equation} \vec{V}\cdot(\vec{V} \times \nabla g) = g\vec{V} \cdot (\nabla\times\vec{V}). \end{equation} The left hand side is zero. Since $g$ is not identically zero, it follows that $\vec{V} \cdot (\nabla\times\vec{V}) = 0$.

  • $\begingroup$ Thank you for your answer! Two things: Why is $g$ not identically zero? And why is the left hand side zero? I appreciate any sort of help I can get :) $\endgroup$ – Kam Oct 20 '19 at 15:23
  • $\begingroup$ If $g$ is zero, so will be $gV$ and such a function will not be terribly interesting to examine further. Left-hand size is zero because $\vec{V}\cdot(\vec{V} \times \nabla{g}) = (\vec{V} \times \vec{V})\cdot\nabla{g} = 0\cdot\nabla{g} = 0$. $\endgroup$ – Amey Joshi Oct 20 '19 at 15:25
  • $\begingroup$ Thank you! I would upvote but they won't allow it before I have a 15 reputation :P $\endgroup$ – Kam Oct 20 '19 at 15:27
  • $\begingroup$ Never mind. I am happy that your problems is solved :). $\endgroup$ – Amey Joshi Oct 20 '19 at 15:28

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