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 > 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.

