```Date: Nov 21, 2013 11:23 AM
Author: David C. Ullrich
Subject: Re: Proof that mixed partials commute.

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 hypothesisthat the partials are continuous?>
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