
Re: Proof that mixed partials commute.
Posted:
Nov 21, 2013 2:25 AM


Am 19.11.2013 00:45, schrieb Hetware: > 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}] > > 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? I feel as though I am trying to prove the > obvious, but it is not obvious how to prove it. >
Since you mention the existence and continuity for all partial derivatives up to second order you have to use the equicontinuity as a condition that limits of continuous functions are continuous.
Equicontinuity means exactly what you have written implicitely, namely that your epsilon/delta_functions_ of the other variable are constants, independent of the free other variables.
Generally
lim 1/h( f(x+h,y)+f(x,y))
is a continuously indexed limit of functions in the free variable y.
A general limit of continous/differentiable functions may develop singularities as lim_n>oo (x^n) because for every n you need another epsolon/deltabound.
The commutativity of independent differential operators is at the heart of functional analysis with the rules
d^2 =0 for exterior differential forms or
f_xy  f_yx=0 for the algebraic setting
or rot grad f=0
for the vector analysis setting. So its a much better learning strategy to look for smoothness definitions in the contexts of function spaces in contrast to painfully filtering a general sufficient condition for the interchageability of pointwise limits for real functions in the supremum norm.

Roland Franzius

