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.
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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)
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)
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)
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)
Why is it called solenoidal? -- Abdull 13:46, 12 September 2007 (UTC)
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)
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
on the domain
This field can be shown to be divergence-free on this domain. However, the net flow through a sphere S contained in is .
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.
InfoBroker2020 ( talk) 11:05, 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 H2dR(U)=0). It is false even in some simply connected domains, like R3-{0}. -- Txebixev ( talk) 15:36, 20 February 2014 (UTC)
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)
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)
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.
This
level-5 vital article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||
|
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)
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)
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)
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)
Why is it called solenoidal? -- Abdull 13:46, 12 September 2007 (UTC)
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)
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
on the domain
This field can be shown to be divergence-free on this domain. However, the net flow through a sphere S contained in is .
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.
InfoBroker2020 ( talk) 11:05, 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 H2dR(U)=0). It is false even in some simply connected domains, like R3-{0}. -- Txebixev ( talk) 15:36, 20 February 2014 (UTC)
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)
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)