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

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Robin Chapman

Posts: 40
Registered: 10/29/12
Re: Proof that mixed partials commute.
Posted: Nov 20, 2013 3:22 AM
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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.

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