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

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Proof that mixed partials commute.
Posted: Nov 18, 2013 9:17 PM
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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}]


f_x(x,y) = lim(h->0) (f(x+h, y) - f(x,y))/h
f_xy(x,y) = lim((k->0) lim(h->0)
. . (f(x+h, y+k) - f(x+h, y) - f(x, y+k) + f(x,y))/hk

f_yx(x,y) = lim((h->0) lim(k->0)
. . (f(x+h, y+k) - f(x, y+k) - f(x+h, y) + f(x,y))/hk

The essence of the proof is for what properties does
lim(x->a) lim(y->b) g(x,y) = lim(y->b lim(x->a) g(x,y).

Is continuity of g(x,y) sufficient?

After that, does the continuity of f_x, f_y, f_xy and f_yx suffice to show
. . g(h,k) = (f(x+h, y+k) - f(x, y+k) - f(x+h, y) + f(x,y))/hk
is continuous for all x,y?

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