
Re: Proof that mixed partials commute.
Posted:
Nov 21, 2013 11:23 AM


On Wed, 20 Nov 2013 19:27:11 0500, Hetware <hattons@speakyeasy.net> wrote:
>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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