
Re: Proof that mixed partials commute.
Posted:
Nov 22, 2013 11:57 PM


On Tuesday, November 19, 2013 3:59:38 PM UTC8, Hetware wrote: > On 11/19/2013 11:17 AM, dullrich@sprynet.com wrote: > > > 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] > > > > I guess I should have included the intermediate steps. I had intended > > that the order of taking limits should be ambiguous. > > > > >> > > >> 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. > > > > Suppose that x(t)=at+c and y(t)=bt+c where a and b are arbitrarily > > chosen real number constants where at least one is not zero. Is it true > > 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] > > > > for all possible a and b? Assuming F and G are continuous in the > > neighborhood of {x(c),y(c)}. Is there any case of Limit[F(x), x>c] > > Limit[G(y), y>c] not covered by the set of all pairs {{a,b}}?
Where are you bringing out the squares? All I see is to establish the relative values of variables, in limits, is a process, that maintains that relative scale, in limits, or bounds.
(And bounds don't.)
I am expecting that when you mutually analyze the codifferential for x and y back in the fundamental theorem, it is to squares here that later it is as to approximations. Then, when you go to exchange limits of them are they share, sure that's a conserved notion that the order is immaterial except here for example where its ranged over the bounds, and simple enough order of integration is in effect. (Here of perfect distance.)
Here a line is a circle to some infinity distant point, where: to be infinitely distant, that point is to the line, infinitely distant from each of its points. Then, the other point might be closest to another point on the line, as the next definition of geometry, for a ray from that point to infinity (eg as through its vector, in real numbers). That's simply not an unusual expectation where then that the order of integration as effective, is on that line, for the perspective and projection of some actual _ordering_ of the integration of components, that are mutual, an actual ordering itself of the integration of components, here establishes a square. The integration of components could be of any other, and, for a particular action of the components, is direct to that, with usual notions of conservation and symmetry.
So, when you want the variables to share the limit, or the variable in the one sense to be limited by another, here in the standard framework that y and x are interchangeable in ordering, as long as their own identity is maintained, they're not just interchangeable in the ordering, it's that due the natural symmetries of Lebesgue and Riemann integrals, in the Cartesian, the components of integration are as to squares. To actually estimate that effect, instead of their own vector model, of actions, the effective integration of the order of the components of integration, would be effective terms in the relative estimation effect, of even maintaining x and y as vector bases: of integrable components.
Here, the declaration and notation, while direct, in relative x and y in their limits as a natural monotonic process, working up how the integration is an integration of the integration components, that entire families of usually regular analytical frameworks of those components are as to that they would be more concisely expressed in different analytical components and in their relation concise, those would add up. So, what a system has already been constructed to be integrating it, then integrating over that (Which is refinement) would naturally fall to lines. It's like: Achilles and the tortoise are racing from A to B. As usual, they race directly from A to B. It is not the concern that Achilles could wait the entire race and win at the last moment. (This could be however far it is.) This is usually framed as in a line or a racetrack which is a quadrilateral with semicircle track segments on the sides. Starting behind, Achilles could be always back in time, but starting the same, he always beats the turtle at every race, it is established here from their rates of motion and the simple enough rules of motion as their numeric equations, classically (phys.), that he could start far enough behind, at any race, and win. That there is a race the turtle could win, is defined here from its constant rate progress of Achilles, A. Simply enough as A(t) and T(t) are functions of their distance, _only between and through A and B_, then the track is an effective definition of a circuit.
Then, the time the Turtle could win, compared to what Achilles could save waiting to start the race, are different in whatever is A(t), that A_max(t) = T(t). This is that whatever the race, Achilles could wait the entire beginning or the race T(t)  A(t), or (TA)(t). (Rather, BA , that the minimum time Achilles can win the race, is the maximum time the Turtle or Tortoise has to win the race. ) If the turtle gets close enough to the finish line then it could win, then if it can see Achilles start, it can wait as long as it takes to get going again, that is the time the turtle can win.
Regards, Ross Finlayson

