Hetware
Posts:
148
Registered:
4/13/13


Re: Proof that mixed partials commute.
Posted:
Nov 19, 2013 6:59 PM


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}}?

