Symmetry of higher order partial derivatives

In summary, the discussion was about Clariut's theorem and its application to mixed derivatives. The question was raised about an example on Wikipedia where the derivatives were not continuous and not equal. It was suggested to check the talk page for a more detailed explanation, and it was mentioned that there are often ugly counterexamples in multivariable differentiation theory. The discussion also touched on the importance of understanding these concepts in order to progress to more beautiful areas of math.
  • #1
saminny
9
0
Hi,
As per Clariut's theorem, if the derivatives of a function up to the high order are continuous at (a,b), then we can apply mixed derivatives. I am looking at
http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives

and I cannot understand in the example for non-symmetry, why the derivatives are not continuous and not equal. Can someone please explain that example?

thanks,

Sam
 
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  • #2
Did you also check out the talk page of the Wikipedia article you linked?

It is explained in more detail there.
 
  • #3
Yeah that counterexample is basically one huge ugly computation that you could verify yourself. Unfortunately there are a wealth of such ugly counterexamples in multivariable differentiation theory because there are no nice basic theorems. The only theorems I would consider nice are the inverse function theorem and the implicit function theorem. Unfortunately you have to expose yourself to this ugliness if you want to understand more beautiful areas of math such as complex analysis, harmonic analysis and PDE's, etc.
 

What is the concept of symmetry of higher order partial derivatives?

The symmetry of higher order partial derivatives refers to the relationship between the mixed partial derivatives of a function. It states that if a function has continuous second-order partial derivatives, then the order of differentiation does not matter and the mixed partial derivatives will be equal.

Why is symmetry of higher order partial derivatives important in mathematics?

The concept of symmetry of higher order partial derivatives is important in mathematics because it simplifies the process of finding second-order derivatives. It also allows for easier calculation of critical points and determination of whether they are local maxima, minima, or saddle points.

How is the symmetry of higher order partial derivatives demonstrated mathematically?

The symmetry of higher order partial derivatives can be mathematically demonstrated using the Clairaut's Theorem. This theorem states that if a function has continuous second-order partial derivatives, then the mixed partial derivatives are equal at a given point.

What are the practical applications of symmetry of higher order partial derivatives?

The symmetry of higher order partial derivatives has numerous practical applications in fields such as physics, engineering, and economics. It is used to analyze the behavior of systems and to optimize functions in these fields.

Are there any exceptions to the symmetry of higher order partial derivatives?

Yes, there are some exceptions to the symmetry of higher order partial derivatives. These occur when the function does not have continuous second-order partial derivatives or when the function is not defined on a closed region.

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