Talk:Symmetry of second derivatives

message: To convince a skeptical reader, at least one counterexample should be included in the following paragraph, or the paragraph should be deleted.
In most "real-life" circumstances the Hessian matrix is symmetric, although there are a great number of functions that do not have this property. Mathematical analysis reveals that symmetry requires a hypothesis on f that goes further than simply stating the existence of the second derivatives at a particular point. Schwarz' theorem gives a sufficient condition on f for this to occur.
--HopingToBeUseful (talk) 16:36, 3 May 2016 (UTC)
thread_title: More on counterexample

message: 131.111.16.20 (talk) 16:20, 18 May 2015 (UTC)
thread_title: The link (1) is broken

message: I think it would help, for the counterexample, to check whether it is locally integrable, at (0,0). Charles Matthews 22:39, 14 January 2006 (UTC)
thread_title: Counterexample

message: I'm very curious, as I know this property as "Young's Theorem," but this is not mentioned by name here.  (dphrag 07:19, 14 June 2006 (UTC)
replies: text: Yeah, I also heard it is called that way. I mentioned this in the article. Oleg Alexandrov (talk) 01:01, 15 June 2006 (UTC)
thread_title: Name

message: [Could someone write up such an example here or in its own article, and add that to the list of mathematical examples?]
I removed it. --M1ss1ontomars2k4 (T | C | @) 21:48, 22 June 2006 (UTC)
thread_title:  Copied out from article 

message: Definitely agree that this should be merged with Clairaut's theorem. --Macrakis 01:33, 1 June 2007 (UTC)
thread_title: Merger

message: I find it a bit confusing at first to read about a counterexample directly after a theorem (if there is a counterexample, it's not a theorem). However, the counterexample relates to the naive idea about symmetry of the Hessian where continuity is disrespected. Maybe the paragraph could be reformulated like: "If the condition of continuity of the second derivates is dropped, counterexamples with unsymmetric Hessian are possible". I think that would make it clearer that this is -- of course -- no counterexample to the preceding theorem, but only to the more naive idea. 134.169.77.186 (talk) 08:36, 8 April 2009 (UTC)
thread_title:  Counterexample 

message: Alongside the 3d plot of the counterexample function, I think a 3d plot of its second derivative might be helpful. The piecewise function is continuous, and the first derivatives just about are as well (perhaps having at worst only removable singularities) but the second derivative is plainly discontinuous. It limits to different values (ranging between +/-1) according to the direction from which the origin is approached. (This seems to make it easier to understand why the actual value at that point, or even whether it is undefined, should depend on subtleties of how the derivatives are defined, and particularly on the direction in which the limits are calculated.)  Cesiumfrog (talk) 02:38, 12 March 2013 (UTC)
thread_title:  Image 

message: What does the "Clairaut's constant" section have to do with the topic of this page?  Either this should be explained, or it should be moved elsewhere.  McKay (talk) 03:53, 28 April 2009 (UTC)
thread_title:  Clairaut's constant 

message: Just want to add my two cents - I think the language should be changed so that the theorem is correct, perhaps by adding a smoothness assumption, or by assuming that the "total derivative" / "Jacobian" exists  — Preceding unsigned comment added by 199.38.133.55 (talk) 20:25, 7 March 2012 (UTC)
thread_title:  Always Equal? 

message: Should there be a note about the non-symmetry of total derivatives? 
For example, in physics relating acceleration to the velocity as a function of distance:
:\ddot x\equiv\frac{d\dot x}{dt}=\frac{dx}{dt}\frac{d\dot x}{dx}=\dot x\frac{d\dot x}{dx}
=\dot x D_x D_t x \neq \dot x D_t D_x x=0
Cesiumfrog (talk) 01:55, 11 March 2013 (UTC)
thread_title:  total derivatives 

message: This topic has so many names, perhaps there needs to be a history section to explain it?
*Clairaut-Schwarz's theorem
*Clairaut's theorem (presumably published in french someplace 1713-1765 and one of several of Clairaut's theorems).
*Schwarz's theorem (or Schwarz integrability condition?) (maybe published where it had more influence on germanic language authors 1843-1921)
*Euler’s Theorem for mixed derivatives. Apparently, "Euler first published the Mixed Derivative Theorem in 1734, in a series of papers he wrote on hydrodynamics." 
*Young's theorem (W.H. Young ~1910? Or is it one coauthored with another Young?)
Funnily enough, there seems to be a uni math course that deals directly with the historical sublety here. Also. Seems they each may have proved something very slightly different.
Cesiumfrog (talk) 22:47, 12 March 2013 (UTC)
thread_title:  Naming and history 

message: might be useful to mark the comments that refer to totally obsolete texts?
the following is trivial but perhaps interesting
let F(x,y)=int dxdy f(x,y), then d/dy(dF/dx) is the limit of f
along y and d/dx(dF/dy) that along x
because provides a construction of a pathology (by the way, I
would have employed this word instead of counterexample avoiding
the discussion above). I leave to someone more skilled than me
to consider if this allows interesting conclusions on the size
of the pathological domains.
in the section sufficiency of twice-differentiability, 'all partial derivatives' is 'all FIRST partial derivatives?'
 pietro2A00:1620:C0:64:21C:61FF:FE03:A4C (talk) 13:12, 28 March 2014 (UTC)
thread_title:  two comments 

message: Looking at the history section there seems to be various conditions used. Some questions:
# In  Clairaut first attempted proof what were the conditions
# What condition was used by Schwarz
# Dini strengthen the conditions, how?
# What were the Hobson and Young conditions?
Later down in the Sufficiency of twice-differentiability section we have the condition that 
:that all partial derivatives are themselves differentiable.
but it does not say who's result this is. The section also notes a strengthening of the theorem due to Peno but this does not seem to be mentioned in the history. 
Is there a way we can be a bit more precise about who did what when? --Salix alba (talk): 05:33, 26 August 2020 (UTC)
replies: text: The historical account of Higgins can be found by clicking on the wayback link in the article. I don't think there's much more detail for most of the questions you've asked. Rudin's book on Principles of analysis gives various conditions and proofs. There is also the 1907/1921 book of Hobson. Dini, Schwarz, Young and Peano appear in the index. And the counterexample of Peano comes from Hobson and Apostol. Hope that helps. Mathsci (talk) 07:30, 26 August 2020 (UTC)
replies: text: See also this. Mathsci (talk) 07:39, 26 August 2020 (UTC)

text: I strongly agree with the issues raised by Salix Alba. The historical account given in the article is a good starting point, but some further information would improve it a lot. At the moment, it seems that the best statement and proof are (as usual) those by Giuseppe Peano. His statement has a stronger content (it gives the existence of the other mixed partial derivative for free!), and the proof is even simpler than the one reported in the article. Further  results by Hobson or Young may be even better. In any case, I think this wiki article should pinpoint the best statement and the best proof --and maybe give the right credit to the authors. pma 18:06, 19 October 2021 (UTC)
thread_title:  History and conditions 

message: this is equivalent to the existence of the second (frechet) derivative, right? in this form its called dieudonne theorem, maybe this should be mentioned. Peter Grabs (talk) 19:39, 26 May 2024 (UTC)
thread_title:  sufficiency of twice-differentiability