Existence and Mixed derivatives

  • #1
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Main Question or Discussion Point

I remember before reading bits and pieces about how if we have a function of two variables, say f = f(x,y), then it must be true that d/dx(df/dy) = d/dy(df/dx), where the "d"'s are partials.

Can anyone guide me to what this theorem is called or to its implications? Also, does it work in reverse? i.e. if it is true that d/dx(df/dy) = d/dy(df/dx) for some function f, then does f necessarily exist?
 

Answers and Replies

  • #2
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This result is called Clairaut's theorem, and it merely requires that all the second partial derivatives are continuous. The reciprocal of this theorem is not true, since there is at least one function with a pair of 2nd partial derivatives equal at a point while at least one of the 2nd derivatives is not continuous at that point.
 
  • #3
HallsofIvy
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I remember before reading bits and pieces about how if we have a function of two variables, say f = f(x,y), then it must be true that d/dx(df/dy) = d/dy(df/dx), where the "d"'s are partials.
Provided the second partials are continuous.

Can anyone guide me to what this theorem is called or to its implications? Also, does it work in reverse? i.e. if it is true that d/dx(df/dy) = d/dy(df/dx) for some function f, then does f necessarily exist?
If f does not exist then what in the world would you mean by "some function f"? Have you miswritten?
 
  • #4
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Yes, I see now how I wrote doesn't make sense. I was trying to use this fact to solve the problem in my last post of the "Expressing multi-variable functions" Thread, but I guess I can't use this fact to check for whether a function exists or not.
 

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