
Re: Proof that mixed partials commute.
Posted:
Nov 18, 2013 9:17 PM


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
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?

