A question about mixed partial derivative

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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 ?
 

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  • #2
selfAdjoint
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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.
 
  • #3
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Thanks for the reply.

Does that mean that double differentiability implies the fxy=fyx equality?(Or reverse plus continuity of other partial derivatives?)
 
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selfAdjoint
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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.
 
  • #5
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Thanks again!
This really clarifies some things...
 

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