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My question is about the proof of how the Biot-Savart law can be used for vortex filaments. This is what I have in my textbook: enter image description here

However, I'm not certain how it was derived - I understand that the Biot-Savart law originally came from electromagnetic theory, but could anyone present the proof for it's use for vortex filaments?

Intuitively though, something seems off - a vortex filament is just made out of lots of individual vortexes put together in a filament, so shouldn't the induced velocity at a point P of an infinitesimal section of the filament be zero unless $d\vec{l}$ and $\vec{r}$ are perpendicular... but clearly it's not the case, $$ d\vec{V} = \frac{ d\vec{l} \times \vec{r} }{ \lvert \vec{r} \lvert ^3} $$ , which is nonzero even when $d\vec{l}$ and $\vec{r}$ aren't perpendicular

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    $\begingroup$ I think this might be a better fit for physics.stackexchange. The vortices may be related to aviation, but proving a theorem is better suited for physics. $\endgroup$ – Bianfable Oct 25 '19 at 16:22
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    $\begingroup$ The whoda whatzit law for vortex what-the-hecks? Over my head... $\endgroup$ – John K Oct 25 '19 at 16:37
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Why would

the induced velocity at a point P of an infinitesimal section of the filament be zero unless $d\vec{l}$ and $\vec{r}$ are perpendicular

?

The cross product in $$ \frac{ d\vec{l} \times \vec{r} }{ \lvert \vec{r} \lvert ^3} $$

does not change signs when going over the $\frac{\pi}{2}$ angle in between vectors, so the infinitesimal filament sections $d\vec{l^-}$ (immediately before $d\vec{l}$) and $d\vec{l^+}$ (immediately after $d\vec{l}$) have contributions to $d\vec{V}$ that do not cancel each other out. Therefore, $\vec{V}$ is not just a function of a single filament section, but of the entire filament, hence the integral.

If you check the next section in the source of your image, titled Straight-vortex case you will see this in action.

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Nevermind guys, I finally found a proof for this, after a long time of googling:

https://www.whoi.edu/fileserver.do?id=218347&pt=10&p=116694

Basically, you use the differential definition of vorticity (w=curl of velocity) and the incompressibility condition (divergence of velocity = 0), then combine them and solve.

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    $\begingroup$ Can you please summarize the results into your answer? Otherwise, it defies the purpose of SE. $\endgroup$ – JZYL Oct 28 '19 at 20:03

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