On Wed, 20 Nov 2013 06:55:30 -0500, Hetware <email@example.com> 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.