Existence and Mixed derivatives

  • #1
224
0
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
1,250
2
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
Science Advisor
Homework Helper
41,833
961
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
224
0
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.
 

Related Threads on Existence and Mixed derivatives

  • Last Post
Replies
3
Views
2K
Replies
4
Views
766
Replies
4
Views
575
Replies
1
Views
1K
Replies
4
Views
2K
  • Last Post
Replies
2
Views
2K
Replies
4
Views
4K
Replies
7
Views
1K
Replies
8
Views
5K
Replies
4
Views
7K
Top