Talk:Solenoidal vector field

Is there any reason at all not to edit this down so that it just says (1) solenoidal means zero divergence and (2) this is equivalent to having a vector potential? I don't think there's any other content on this page that isn't either incorrect or irrelevant. Perhaps it's worth adding that incompressible fluid flow => a velocity field with zero divergence.

Much of it should go methinks

True, most of the article seems to go on and on and the essence of the subject gets lost. What I find odd is that the article starts by saying that a solenoidal vector field is when div v = 0 and ends by saying that [...] is why div v = 0. Maybe mathematics is a tautology but that's too much for me. A rewrite (leaving out the talk of cars highways and gears) would be good thing. --12:27, 16 Mar 2005 (UTC)

OK: done, rather brutally. Gareth McCaughan 14:11, 2005 Mar 20 (UTC)

While here - there is certainly a connection with the Poincaré lemma. Now, what is this page trying to achieve? Is it going to state the lemma, in effect, in vector calculus terms, so as not to frighten the horses? Is it going to state some valid special case? Is it going to try to prove (better, sketch a proof of) something? Anyway these points seem bound up with trying to get our necessary and our sufficient conditions clearer.

Charles Matthews 17:03, 20 Mar 2005 (UTC)

My recent reversion

I just reverted an anon contribution, and hit "Return" before finishing my comment. I meant to say that the proof in the article is rigurious. Even if the curl is not the same as the cross-product, they obey the same laws. Oleg Alexandrov (talk) 22:55, 13 November 2005 (UTC)

B is divless not H

I notice that Special:Contributions/83.131.29.96 has changed magnetic flux density to magnetic field in a number of articles. It's clearly incorrect in this case (I reckon), so I've reverted it. Please discuss the matter here if you disagree. Also, it'd be helpful if you registered a user name. Thanks, --catslash 17:04, 7 July 2007 (UTC)

What does "divless" mean? Just as electric flux is integral of E over an area, so is magnetic flux integral of B over an area, and just as E is an electric field, so is B an magnetic field. H is just an magnetic analogue of electric displacement field D, and it is not itself an magnetic field, just as D is not itself an electric field. Flux density of an field is the field itself ("Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density."), and magnetic flux density redirects to magnetic field for more than a year anyway. --83.131.8.91 10:46, 8 July 2007 (UTC)

Actually B is now considered the Magnetic Field by physicists, so please never use Magnetic Field to mean H. Special Contributions 83.131.29.96 is totally correct on that one! — Preceding unsigned comment added by Jwkeohane (talk • contribs) 00:34, 17 June 2014 (UTC)

Name

Why is it called solenoidal? --Abdull 13:46, 12 September 2007 (UTC)

Simply connected domain

div f = 0 => F such that f = curl F is only supposed be true on specific domains (I'm not sure whether it's star-like or simply connected one implying the other anyway). Don't you think it should be added ? —Preceding unsigned comment added by 81.245.62.57 (talk) 08:48, 21 May 2008 (UTC)

Correctness of definition?

As far as I understand the definition of a vector field that is solenoidal on a domain Omega (using the term domain loosely here) is that there is no net flow through any closed surface (again using that term loosely) contained in Omega. For many types of domains, such as e.g. simply connected or star-shaped this is the case (for a continuously differentiable field) if and only if the field is divergence-free.

However, this analogy does not hold in all cases. A typical example, that has some relevance e.g. in the Earth Sciences, are annuli and thick spherical shells. For such domains a field can by divergence-free, but non-solenoidal. An example is given by the field

$v(r)={\frac {r}{|r|^{3}}}$

on the domain

$\Omega _{(\alpha ,\beta )}={\Big \{}r\in \mathbb {R} ^{3}\,{\Big |}\,0<\alpha <|r|<\beta {\Big \}}$

This field can be shown to be divergence-free on this domain. However, the net flow through a sphere S contained in $\Omega _{(\alpha ,\beta )}$ is $4\pi$.

This does not contradict the divergence theorem, since the sphere S does only constitute one part of the boundary of the volume enclosed by it, the other one being the inner boundary of the domain itself.

See e.g.

• Carsten Mayer, Lecture Notes in Geomathematics
• G. Backus, Poloidal and Torroidal Fields in Geomagnetic Field Modelling, Reviews of Geophysics, no. 1, vol. 24, 1986, pp. 75-109

InfoBroker2020 (talk) 11:05, 28 December 2009 (UTC)

Flicking through a few books, I find solenoidal most commonly defined as divergenceless - however theses are largely elementary texts which confine their attention to R³. Michiel Hazewinkel "Encyclopaedia of Mathematics" vol. 9 p402 (viewable in Google Books) gives a stronger surface-integral definition, and notes that divergencelessness does not imply solenoidality in domains with holes. I can't immediately find any other book giving this stronger definition.

We should follow the sources; start with the 'elementary' definition, then follow with the 'advanced' definition, noting how they differ. --catslash (talk) 17:53, 28 December 2009 (UTC)

Calling solenoidal the divergengeless (or incompressible) vector fields is misleading. The term solenoidal should be restricted to vector fields having a vector potential. Solenoidal implies divergenceless, but the converse is true only in some specific domains, like R³ or star-shaped domains (in general: domains U having H²_dR(U)=0). It is false even in some simply connected domains, like R³-{0}. --Txebixev (talk) 15:36, 20 February 2014 (UTC)

Terrible introduction

The introductory section includes this sentence:

"A common way of expressing this property is to say that the field has no sources or sinks."

No, that is most definitely not a common way of expressing the property of a vector field's having zero divergence. It is not a way at all. It is not true, as the simplest examples show.50.205.142.35 (talk) 10:35, 31 December 2019 (UTC)

What counter-examples had you in mind? catslash (talk) 13:08, 31 December 2019 (UTC)

There is no way solenoidal comes from Greek "constrained as if in a pipe"

These fields are almost certainly called solenoidal because of their relationship to solenoidal magnetic fields (i.e. magnetic fields generated from pipe-like coils of current-carrying wires). This seems to me to be a folk etymology (and a quite bizarre one at that), it definitely needs a source. INLegred (talk) 21:24, 20 February 2023 (UTC)

Wiktionary says the word comes from Ancient Greek σωληνοειδής "pipe-shaped"; Liddell & Scott's dictionary confirms that such a word was used in Ancient Greek. How it came to be applied to the vector field I don't know, though your explanation does seem rather more plausible. - LaetusStudiis (talk) 00:18, 23 March 2023 (UTC)

It's "solenoidal" because of the curl. You can imagine each vector in the field as being a tiny pipe, and the pipes must always be connected so that there's just as much coming in as there is going out: the pipes don't leak. So instead of the conventional "combing hair without flat spots or peaks", think "combing pipes so they're always connected". I'm guessing this is a 19th century idea. If you read James Clerk Maxwell's diaries, they're filled with these esoteric thoughts of mechanistic fluids sloshing around in cycles. The concept of a vector field, clearly popular in the 19th cent, its hard to imagine that its much older than that. 67.198.37.16 (talk) 16:23, 22 March 2024 (UTC)