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

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 Robin Chapman Posts: 412 Registered: 5/29/08
Re: Proof that mixed partials commute.
Posted: Nov 19, 2013 3:00 AM
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On 19/11/2013 02:17, William Elliot wrote:
> On Mon, 18 Nov 2013, Hetware wrote:
>

>> In _Calculus and Analytic Geometry_ 2nd ed.(1953), Thomas provides:
>>
>> 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.
>>
>> Starting from the definition of partial differentiation, and using the rules
>> of limits, along with a modest amount of basic algebra, I came up with this:
>>
>> f_x(x,y) = Limit[[f(x+Dx,y)-f(x,y)]/Dx, Dx->0]
>>
>> f_yx(x,y) = Limit[[f_x(x,y+Dy)-f_x(x,y)]/Dy, Dy->0]
>> = Limit[
>> [[f(x+Dx,y+Dy)-f(x,y+Dy)]-[f(x+Dx,y)-f(x,y)]]/DyDx
>> , {Dy->0, Dx->0}]

>
> Aggg!Crampedcomputertalk,ugh,ugh.Don't

He must have shares in Wolfram Enterprises :-)

Date Subject Author
11/18/13 Hetware
11/18/13 William Elliot
11/18/13 Hetware
11/19/13 Robin Chapman
11/19/13 David C. Ullrich
11/19/13 Brian Q. Hutchings
11/19/13 Hetware
11/20/13 Robin Chapman
11/20/13 Hetware
11/20/13 David C. Ullrich
11/20/13 Hetware
11/20/13 Ki Song
11/21/13 Robin Chapman
11/21/13 David C. Ullrich
11/20/13 Robin Chapman
11/20/13 Hetware
11/21/13 Robin Chapman
11/21/13 Robin Chapman
11/21/13 David C. Ullrich
11/22/13 ross.finlayson@gmail.com
11/21/13 Roland Franzius

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