Talk:Conservative vector field

error in the title of the vector field figure !??!

this is a rotational vector in the figure and not an irrotational as it states !! —Preceding unsigned comment added by 94.249.102.105 (talk) 11:03, 8 May 2009 (UTC)

Indeed. I changed the caption. -- Crowsnest (talk) 12:26, 8 May 2009 (UTC)

the vector field figure appears to be incorrect!??!

If the vector length represents the magnitude, then the figure is wrong i THINK. The vectors near the origin should continue to grow. I suggest that if the vectors near the origin have been shortened for clarity, this should be stated somewhere. It is a very nice figure nevertheless! (prettier than IDL can do)

;; IDL code 
nx = 100  &  ny =100         ; dimensions
x=dblarr(nx,ny)
y=dblarr(nx,ny)
xv=dindgen(nx)/nx            ; x vector [0..99]/100 --> [0..1)
yv=dindgen(ny)/ny            ; y vector
for j=1,ny do x[*,j-1]=xv      ; replicate x vector -> x matrix
for i=1,nx do y[i-1,*]=yv       

; erroneous plot??

fx = -y/(x^2+y^2)
fy =  x/(x^2+y^2)

vel, fx, fy, title='Wiki',len=1   ; a curvy arrow vector randomly sampled field plot

Bdb112 (talk) 07:48, 15 January 2010 (UTC)

Opening sentence

I may be being a bit pedantic, but I'd like to change the first line from:

• In vector calculus a conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.

to

• In vector calculus a conservative vector field is a vector field which is the gradient of a function which is known as a scalar potential.

...or something similar, just to get across immediately what the article is referring to as a scalar potential. — Preceding unsigned comment added by Larryisgood (talk • contribs) 17:37, 29 July 2011 (UTC)

Correction - "connected" should be "simply connected"

Excellent page. Thanks to everyone who contributed. However, I found one error:

The line:

 "(However, in any connected subregion of S, it is still true that it is conservative..."

I believe should say:

  "(However, in any SIMPLY connected subregion of S, it is still true that it is conservative..."

S itself is connected. A connected subregion encircling the Z axis wouldn't do at all. I hope this comment is helpful (and correct!).

Thanks.

 Catgod119  — Preceding unsigned comment added by Catgod119 (talk • contribs) 06:43, 16 January 2013 (UTC) 

Recent edits

This edit has a number of issues making it unsuitable for an encyclopedia. First of all, it is simply wrong to say that "More generally, a conservative field is a natural field which after the field has undergone changes, and been returned in any mode to its original state, the total power done by externals on the field is equal to the total power done by the field supervening external forces." This is also totally unsupported by the source cited in that sentence. The notion of a conservative vector field is well-known in mechanics, and there's no need for such bafflegab. Second, the paragraph on solenoidal vector fields is completely offtopic in the lead, although I would, not be opposed to having a section on solenoidal vector fields, the Helmholtz decomposition, and so forth. Finally, there is an issue of citation abuse. The article does not need references to poor quality sources like mathworld and non-reliable sources like Youtube and Khan Academy. The material is thoroughly standard and can be found in very high quality sources like the book by Marsden and Tromba and the book by Arfen and Weber already referenced in the article. The scientific citation guideline does not recommend large piles of citations for very standard facts. Sławomir Biały (talk) 11:34, 29 August 2013 (UTC)

The citations are to improve the article and provide non-technical references; but as I cannot address all your comments at the present moment [have to run an errand], I'll be back. The vast variety of redirect to this page is a problem too ... and have thought that several needs to be created as stand alone articles [all the bold terms are redirects here]. Do not remove the information and citations. --J. D. Redding 12:39, 29 August 2013 (UTC)

The information that I removed from the lead was a paragraph that you wrote regarding total power. This is an error and is unsupported by the citation you provided, as explained above. I also removed a paragraph from the lead that was irrelevant (it was about solenoidal vector fields and Laplacian vector fields, not conservative vector fields). The citations you added do not meet the rather high standards for inclusion in an encyclopedia. You referenced Youtube videos, self-published sources, Mathworld, entire volumes of journals, and obsolete works from the 19th century. What this is supposed to add to the article, I have no idea. Sławomir Biały (talk) 12:53, 29 August 2013 (UTC)

Apparently you have just reverted the article without trying to improve it. This is a problem. --J. D. Redding 12:42, 29 August 2013 (UTC)

But I did improve on your revision of the article, by removing the poor sources that you added, removing the original research that you added, and adding a paragraph instead on the well-known applications to mechanics that appear in all of the textbooks on the subject. Your response was to re-insert all of the rubbish that you added originally. Sławomir Biały (talk) 12:55, 29 August 2013 (UTC)

Just got back a bit ago. Kinda reticent to even exchange comment with people like you. Your comments show your disrespect and noncooperation ...among other factors [among which edits (beginning in 2009) and userpage]. People of SA's ilk, which you seem to like, made Wikipedia for the worse.

But, against my better judgement [and for people that look at the talk page], ... removing information and citations is not "improving the article" [as a general case] ...

The following are secondary sources ...

• Conservative Vector Fields - The Definition and a Few Remarks - patrickJMT, YouTube
• Conservative Field -- from Wolfram MathWorld. Mathworld.wolfram.com.
• Path Independence for Line Integrals | Line integrals in vector fields. khanacademy.org.
• The Messenger of Mathematics, Volumes 30-31. Macmillan and Company, 1901.
• Introduction to the Calculus of Variations. By William Elwood Byerly.
• Introduction to the theory of analytic function. By James Harkness, Frank Morley. Macmillan and Co., limited, 1898. p210.
• Advanced Calculus. By Edwin Bidwell Wilson. Ginn, 1912. p298
• Irrotational Field -- from Wolfram MathWorld. mathworld.wolfram.com.

... And there is the lines of:

• irrotational : A vector field which the infinitesimal rotation of 3-dimensional space takes the value zero.
• Conservative system.
• $\nabla \varphi$ (Dipole)
• $\varphi$ (phi)

As a last thing ... the multitude of redirects, a issue here, needs to either be: 'split' off as separate articles (eg., irrotational); or, explained and the term made bold.

Anyways, this page needs to include more information and is incomplete as it needs non technical information. --J. D. Redding 01:57, 30 August 2013 (UTC)

I have removed the {{incomplete}} tag, as the article is not missing anything obvious. It needs copyediting, though. Ozob (talk) 14:25, 29 August 2013 (UTC)

I am puzzled by your comment. First, your references on the whole aren't very good. I don't mind adding the Khan academy video in the external links section (since that seems to be the trendy thing to do), but I'm not impressed by the other video, the MathWorld links are junk, and the other links are either irrelevant or not particularly high quality. Second, the article has the definition of irrotational (the curl being zero), and while I guess a separate article could be written it would have significant overlap; the article already mentions conservation of energy; I am not sure what you want to say about dipoles, other than perhaps to use them as an example; and φ is just a Greek letter. So I don't know what you think might be missing. And I am not sure what you mean by the "multitude of redirects", as the article currently has very few. I'd be happy to see this article improved; I'm just not sure how you think it should be done. Ozob (talk) 02:49, 30 August 2013 (UTC)

Are the references to ancient textbooks supposed to be general references or something? I did look at those and they seemed like very poor general references, and they did not seem to be inserted in a way tailored to support any particular statement. For instance, the only thing in "Introduction to the calculus of variations" that I could find was a brief discussion of Hamilton's principle, which is a bit peripheral to this article. As far as I can tell, that text does not even mention vector fields. The Messenger of Mathematics is a research journal, and you seem to be referencing two entire volumes of it. I have no idea what this reference is supposed to be good for. The analytic functions textbook seems to have almost nothing of relevance to this article. I'm not even going to comment on selfpublished sources like Khan and YouTube, or obviously poor quality sources like MathWorld. Sławomir Biały (talk) 13:57, 31 August 2013 (UTC)

References

Conservative vector fields are necessarily irrotational?

This statement seems to rely on the vector field being once partially differentiable in all its components with respect to all the spatial variables (in other words, the curl exists). The requirement that $\nabla f=\mathbf {F}$ says nothing about whether the curl of F exists or not. The identity used, as well as Green's theorem, is naturally only valid when that curl exists. So the question is, must the curl of F necessarily exist for a conservative vector field? I don't think so. Surely there exist multivariate analogs of the Weierstrauss function or some other function, such that if F has components equal to those functions, then it has a once differentiable potential function but no curl (or divergence, for that matter).--Jasper Deng (talk) 05:33, 22 April 2014 (UTC)

Now reading the discussion by @Ozob and Sławomir Biały: above about this article being poorly sourced (which it is), I can clearly see why my argument probably holds water.--Jasper Deng (talk) 05:55, 22 April 2014 (UTC)

It is not clear to me that the partial derivatives of F must exist for curl F to exist. For example, if f(x) = 0 if x₁ < 0, and f(x) = x₁² if x₁ ≥ 0 (x₁ being one of the components of the position vector x), then F = ∇f is continuous everywhere, and curl F = 0 everywhere, keeping in mind that it is defined as the limit of a path integral divided by the area bounded. However, F is not differentiable where x₁ = 0, which simply means that the expression for curl in terms of partial derivatives breaks, nevertheless keeping the answer to the question posed in this thread's heading as "yes", with or without sources. —Quondum 06:23, 22 April 2014 (UTC)

Yes, the requirement that $\oint \mathbf {F} \ \cdot d\mathbf {r} =0$ seems to ensure that $\nabla \times \mathbf {F} =\mathbf {0}$, but the article on the curl does indeed state that the operator is from continuously differentiable vector fields to continuous vector fields. I'm rather sure that it is possible to construct a function whose gradient's curl is nowhere existent, and if not, it would interest me to know why. My textbook uses the partial derivative definition and "derives" the loop integral definition from Stokes' theorem, but all along assuming that the partial derivatives in the curl operator exist.--Jasper Deng (talk) 07:13, 22 April 2014 (UTC)

In general, definitions in terms of integrals, like the definition of curl in terms of integrals around closed loops, are more forgiving of bad functions than definitions in terms of derivatives, like the definition of curl in terms of partial derivatives (which is in some sense the "real" definition, since it manifests it as the exterior derivative). But sometimes one can rescue derivatives by replacing them with weak derivatives instead. That may or may not be the case here (I haven't made any attempt to check) but I wouldn't find it too surprising. Ozob (talk) 13:51, 22 April 2014 (UTC)

I think that we should be careful of relying too closely on chains of definitions in WP – irrotational in terms of curl, curl in terms of integrals/partial derivatives. The concept of irrotationality makes perfect sense in edge cases in that they are unambiguous (unlike some weak derivatives), and continue to satisfy (AFAICT) the statements made of them. The question then becomes one of whether they are sometimes (notably) defined to accommodate the non-differentiable edge cases. I would be unsurprised if various different approaches are used and accepted, some of them trying to be more general and hence more "forgiving", especially if this leads to questions such as posed in this thread having unconditional answers. —Quondum 22:26, 22 April 2014 (UTC)

@Ozob and Quondum: It seems like other mathematicians can't agree on it, even the expert I usually trust the most didn't have a clear answer on it. I think we have to consider whether these two definitions of the curl are equivalent to each other, since we'd really like it to be so, especially without weak derivatives. Perhaps it could be because only six of the entries in the Hessian matrix of f have to exist (of course, in three dimensions)- i.e. only its mixed second partial derivatives. But still I see no reason why these six alone must all exist.--Jasper Deng (talk) 06:31, 27 April 2014 (UTC)

I don't see any real disagreement (though I'm not classifying myself as a mathematician, and Ozob's comment about the exterior derivative bears thinking about). Using partial derivatives only is simpler, but does not deal with several useful cases. Electromagnetic field theory deals with distributions, for example a uniform surface charge density on a sphere. This leads to a non-differentiable EM field (when regarded as functions on a manifold rather than as distributions), but is a very accurate description of an everyday (classical) scenario. A suitably general treatment will include cases such as this. Choice of a set of definitions such that the statement "a conservative vector field is necessarily irrotation" does not hold would be an unnecessary artefact of an inadequate choice of definitions. It would be pointless to point out where this statement might fail when using certain definitions. The article on curl should be modified to remove the restriction to differentiable fields (I do not see it in the few books that I checked). The only sources that I've browsed (e.g. [[1]]) that look reasonably authoritative simply restrict the applicability of the definition of curl in terms of partial derivative to where the field is differentiable. Lack of applicability of a given definition of curl must not be confused with the nonexistence of the curl. —Quondum 16:15, 27 April 2014 (UTC)