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It is physics (force between loops) but the highlighted term is equal to zero and I don't understand it. It is from Stokes' theorem but I don't understand it.

Mary Dona
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1 Answers1

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$\int \vec{dl}.\nabla f = \int\int curl(\nabla f)da =0 $ since the curl of a gradient is always $0$. Here I used $f$ for $\dfrac{1}{r_{12}}$

Leonid
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  • if i understand, F=r (vector)/r^3, F=∇f, where f=1/r, F is conservative, so ∇xF=0, so the ∇(∇f)=0 – Mary Dona Jul 15 '22 at 15:28
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    Yes indeed. Though in your last equation it should be the curl of gradient of $f$ is zero; your equation makes it seem like you are taking the gradient of the gradient of f. – Leonid Jul 15 '22 at 15:38
  • yes, do you think that ∇2(there is in the picture) is equal to ∇X(∇f)?? Thank you a lot! – Mary Dona Jul 15 '22 at 15:47
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    No that's something different. Notice that you have two sets of variables here $\vec{r_1}$ and $\vec{r_2}$. The notation $r_{12}$ means $r_{12}=|\vec{r_1}-\vec{r_2}|$, hence the notation $\nabla_{2}$ means you take the gradient with respect to $\vec{r_2}$. – Leonid Jul 15 '22 at 16:27
  • so a good explanation is that exist f that is F=∇f, so F is conservative, so ∇X(∇f)=0, so the integral (highlighted term) is equal to zero ?? – Mary Dona Jul 15 '22 at 16:47
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    yes that is correct. – Leonid Jul 15 '22 at 18:00