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

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Hetware

Posts: 148
Registered: 4/13/13
Re: Proof that mixed partials commute.
Posted: Nov 18, 2013 10:13 PM
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On 11/18/2013 9:17 PM, 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



You lost me. Most of what I wrote was not "computer talk". It was
abbreviated mathematical formalism.

This is almost the Mathematica expression for a limit. I should
probably be more pedantic and use square brackets for all arguments.

Limit[x(t), t->c] = Limit[y(t), t->c] = c

Proper Mathematica code is:

Limit[x[t], t->c] = Limit[y[t], t->c] = c

This is not "computertalk". In words (with a bit if refinement added):

|F(x)-F(c)| < epsilon_F ==> 0 < |x-c| < delta_F exists

Given some epsilon_F such that epsilon_F is greater than the absolute
value of F(x)-F(c), there exists delta_F such that the absolute value of
x-c is less than delta_f and greater than zero.

So, reading in that sense (with added the refinement):

|F(x)-F(c)| < epsilon_F ==> |x-c| < delta_F exists

|G(y)-G(c)| < epsilon_G ==> |y-c| < delta_G exists

|x(t)-c| < epsilon_x ==> |t-c| < delta_x exists

|y(t)-c| < epsilon_y ==> |t-c| < delta_y exists


'Now epsilon_F ==> delta_F can be used as delta_F = epsilon_x which
implies delta_x > |t-c| exists.'

Expanded: Now the fact that the positive real number epsilon_f implies
the existence of some delta_F as defined above, delta_F can be used as
the value of epsilon_x.

'So

Limit[F(x(t)), t->c] = Limit[F(x), x->c], etc.

It follows that

Limit[F(x(t)) G(y(t)), t->c]
= Limit[F(x(t)), t->c] Limit[G(y(t)), t->c]
= Limit[F(x), x->c] Limit[G(y), y->c]'

The limits which were previously expressed as functions of two
variables, x and y are now expressed as functions on one variable, t,
without loss of generality. There is no need to reproduce the
individual proofs of the algebraic rules for limits, since such the
results of those rules can be expressed as functions of one variable.






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