The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Proof that mixed partials commute.
Replies: 20   Last Post: Nov 22, 2013 11:57 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 148
Registered: 4/13/13
Re: Proof that mixed partials commute.
Posted: Nov 20, 2013 6:55 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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.

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.