
Re: Proof that mixed partials commute.
Posted:
Nov 19, 2013 11:17 AM


On Mon, 18 Nov 2013 18:45:53 0500, Hetware <hattons@speakyeasy.net> 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. > >Both Thomas and Anton (1980) provide rather longwinded proofs of this >theorem. These proofs involved geometric arguments, auxiliary >functions, the meanvalue theorem, epsilon error variables, a >proliferation of symbols, and a generous helping of obscurity. > >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_xy(x,y) = Limit[ >[[f(x+Dx,y+Dy)f(x+Dx,y)][f(x,y+Dy)f(x,y)]]/DxDy >, {Dx>0, Dy>0}]
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]
> >The only caveat is that the rules for limits, such as /the product of >limits is equal to the limit of the products/, are stated in terms of a >single variable. For example: > >Limit[F(t) G(t), t>c] = Limit[F(t), t>c] Limit[G(t), t>c] > >Whereas I am assuming > >Limit[F(x) G(y), {x>c, y>c}] = Limit[F(x), x>c] Limit[F(y), y>c]. > >I argue as follows. The statement that x>c as y>c can be formalized by >treating x and y as functions of t such that > >Limit[x(t), t>c] = Limit[y(t), t>c] = c > >F(x)F(c) < epsilon_F ==> xc < delta_F exists > >G(y)G(c) < epsilon_G ==> yc < delta_G exists > >x(t)c < epsilon_x ==> tc < delta_x exists > >y(t)c < epsilon_y ==> tc < delta_y exists > >Now epsilon_F ==> delta_F can be used as delta_F = epsilon_x which >implies delta_x > tc exists. 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] > >Am I making sense here?
Not as far as I can see.
>I feel as though I am trying to prove the >obvious, but it is not obvious how to prove it.

