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Are the following two derivatives same?

  1. May 15, 2010 #1
    1. fxy
    2. fyx

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


  2. jcsd
  3. May 15, 2010 #2


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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).
  4. May 15, 2010 #3


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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.
  5. May 15, 2010 #4


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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


    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.
    Last edited: May 15, 2010
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