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

 Messages: [ Previous | Next ]
 Hetware Posts: 148 Registered: 4/13/13
Re: Proof that mixed partials commute.
Posted: Nov 20, 2013 6:55 AM

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
>
> 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.

Date Subject Author
11/18/13 Hetware
11/18/13 William Elliot
11/18/13 Hetware
11/19/13 Robin Chapman
11/19/13 David C. Ullrich
11/19/13 Brian Q. Hutchings
11/19/13 Hetware
11/20/13 Robin Chapman
11/20/13 Hetware
11/20/13 David C. Ullrich
11/20/13 Hetware
11/20/13 Ki Song
11/21/13 Robin Chapman
11/21/13 David C. Ullrich
11/20/13 Robin Chapman
11/20/13 Hetware
11/21/13 Robin Chapman
11/21/13 Robin Chapman
11/21/13 David C. Ullrich
11/22/13 ross.finlayson@gmail.com
11/21/13 Roland Franzius