Are the following two derivatives same?

  • Thread starter sgsawant
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  • #1
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1. fxy
2. fyx


Are the above 2 derivatives equal, in general. Please explain if you know the answer.

Regards,

-sgsawant
 

Answers and Replies

  • #2
phyzguy
Science Advisor
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Yes, in general if these two derivatives exist and are continuous, then they are equal. This is called "equality of mixed partial derivatives". I have also seen it called Clairaut's theorem, although supposedly it was first proved by Euler (like so much of the rest of mathematics).
 
  • #3
HallsofIvy
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As long as [itex]f_{xy}[/itex] and [itex]f_{yx}[/itex] are continuous in some neighborhood of a point, then, at that point, they are equal.
 
  • #4
Landau
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Even better:

Let U in R^2 be open, [itex]f:U\to\matbb{R}[/itex] partial differentiable w.r.t. both variables, and [itex]D_1f[/itex] partial differentiable w.r.t the second variable. Suppose further that (x,y) is in V, and [itex]D_2D_1f[/itex] is continuous at (x,y). Then [itex]D_2f[/itex] is partial differentiable w.r.t the first variable at (x,y), and

[tex]D_1D_2f(x,y)=D_2D_1f(x,y)[/tex].

i.e. we only need [itex]D_2D_1f[/itex] to exist and be continuous at some point in the interioir, this already implies that [itex]D_1D_2f[/itex] exists at that point and the two are equal.
 
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