When do mixed partial derivatives fxy and fyx equal each other?

[vacuum]

Let there be a function f[x,y]: RxR->R

Is there any connection between the differentiability
(I am not sure that this is the right English term - I meant f[x,y]= a*dx+b*dy +something of smaller order)
and the equality fxy=fyx, where fxy means the derivative of f[x,y] first by y, and than by x ?

 

[selfAdjoint]

Originally posted by vacuum
Let there be a function f[x,y]: RxR->R

Is there any connection between the differentiability
(I am not sure that this is the right English term - I meant f[x,y]= a*dx+b*dy +something of smaller order)
and the equality fxy=fyx, where fxy means the derivative of f[x,y] first by y, and than by x ?

The idicated cross partials have to exist of course. Usually you wouls ensure that by requiring that f be twice differentiable in both variables.

 

[vacuum]

Thanks for the reply.

Does that mean that double differentiability implies the fxy=fyx equality?(Or reverse plus continuity of other partial derivatives?)

 

[selfAdjoint]

Originally posted by vacuum
Thanks for the reply.

I think you have to have continuity of the second derivitives at any point where you want to show the cross partials are equal. Since this is just a feature of the cross partials, i.e. it's always true if you have the above conditions, you can't use it to prove the conditions exist.

Pretty generally, unless you have a fiendishly pathological function f, you can always take the equality of the cross partials for true.

 

[vacuum]

Thanks again!
This really clarifies some things...