Hetware
Posts:
148
Registered:
4/13/13


Re: Proof that mixed partials commute.
Posted:
Nov 20, 2013 7:27 PM


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."

