Date: Nov 21, 2013 11:23 AM
Author: David C. Ullrich
Subject: Re: Proof that mixed partials commute.
On Wed, 20 Nov 2013 19:27:11 -0500, Hetware <hattons@speakyeasy.net>

wrote:

>On 11/20/2013 11:07 AM, dullrich@sprynet.com wrote:

>> On Wed, 20 Nov 2013 06:55:30 -0500, Hetware <hattons@speakyeasy.net>

>> wrote:

>>

>>> On 11/20/2013 3:22 AM, Robin Chapman wrote:

>>>> On 19/11/2013 23:59, Hetware wrote:

>>>>>>

>>>>>> That's very bad notation. It's not one limit, it's the limit of

>>>>>> a limit. Should be

>>>>>>

>>>>>> Limt(Limt(...)[x->c][y->c].

>>>>>>

>>>>>> And now the big question is why

>>>>>>

>>>>>> Limt(Limt(...)[x->c][y->c] = Limt(Limt(...)[y->c][x->c]

>>>>>

>>>>> I guess I should have included the intermediate steps. I had intended

>>>>> that the order of taking limits should be ambiguous.

>>>>

>>>> That's the nub of the matter. Iterated limits need not commute.

>>>> One has to show that in this case they do. Putting in deliberate

>>>> ambiguities in your notation sounds a really bad idea.

>>>>

>>>> Of course there are examples where mixed partials are different,

>>>> so your original argument can't have been valid, since it didn't

>>>> use the necessary hypotheses about continuity of partials etc.

>>>>

>>>

>>> But I added my reason for assuming the limits commute. I expressed a

>>> function of two independent variables as the function of a single

>>> variable and appealed to the limit rules for a function of a single

>>> variable to the result.

>>>

>>> The question is whether that reasoning is valid.

>>

>> It can't be valid, since it "proves" something false!

>> Mixed partials are the same _under_ certain hypotheses.

>> Your proof, if valid, would show that they commute

>> _wiithout_ those hypotheses. And that's not true.

>>

>> Many theorems in analysis amount to showing

>> that some particular two limits commuute.

>>

>>

>

>The starting point for the proof, as stated in the OP:

>"Theorem. If the function w=f(x,y) together with the partial derivatives

>f_x, f_y, f_xy and f_yx are continuous, then f_xy = f_yx."

Where in the supposed proof do you _use_ the hypothesis

that the partials are continuous?

>