
Re: Proof that mixed partials commute.
Posted:
Nov 19, 2013 3:00 AM


On 19/11/2013 02:17, William Elliot wrote: > 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
He must have shares in Wolfram Enterprises :)

