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Topic: Proof that mixed partials commute.
Replies: 20   Last Post: Nov 22, 2013 11:57 PM

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David C. Ullrich

Posts: 3,516
Registered: 12/13/04
Re: Proof that mixed partials commute.
Posted: Nov 21, 2013 11:23 AM
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On Wed, 20 Nov 2013 19:27:11 -0500, Hetware <hattons@speakyeasy.net>

>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?


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