Talk:Symmetry of second derivatives

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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)

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))

Yeah, I also heard it is called that way. I mentioned this in the article. Oleg Alexandrov (talk) 01:01, 15 June 2006 (UTC)

1 Copied out from article
2 Merger
3 Counterexample:
4 Counterexample Again

[edit] Copied out from article

[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. --M1ss1ontom a rs2k4 ^{(T | C | @)} 21:48, 22 June 2006 (UTC)

[edit] Merger

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

Agree. -- Hongooi 17:51, 15 June 2007 (UTC)

[edit] Counterexample:

How is the given function a counterexample? The article asserts that "Then the mixed partial derivatives of f exist, and are continuous everywhere except at (0,0)".

However, (0,0) is not in the domain of the original function. -- Heath 24.127.115.128 15:26, 30 August 2007 (UTC)

I tried to make that clearer. Oleg Alexandrov (talk) 04:21, 31 August 2007 (UTC)

[edit] Counterexample Again

I worked through the second partial derivatives of the counterexample and they seem to be equal; i.e., the "counterexample" isn't a counterexample. I got $\frac{\partial^2f}{\partial x \,\partial y} = \frac{\partial^2f}{\partial y \,\partial x} = \frac{(x+y)(x-y)(x^4+10x^2y^2+y^4)}{(x^2+y^2)^3}$ and my TI-89 confirmed it. Comments? Jonah 02:03, 20 October 2007 (UTC)

The counterexample is correct:

let $f_{xy}(0,0) \stackrel{df}{=} \lim_{h \to 0} \frac{f_x(0,h)-f_x(0,0)}{h} \mbox{ and } f_{yx}(0,0) \stackrel{df}{=} \lim_{h \to 0} \frac{f_y(h,0)-f_y(0,0)}{h}$