Re: Name of Theorem



On Fri, 13 Apr 2007, David Moran wrote:

I am a professional mathematician who already has a math degree, but am
working on a meteorology degree. I was working on homework the other night
and I know there's a theorem for multivariate calculus that says if you take
the partial derivative of a function (say f) w.r.t. x and then y, that's the
same as taking the partial w.r.t y and then x. However, I can't remember the
name of the theorem. I want to say it's Clairaut's Theorem, but I can't
remember for sure and I can't seem to find anything regarding Clairaut's
Theorem and partials.

Thanks,
Dave

The name is clear - Clairaut's, as well as Schwarz's.
Without some assumptions of continuity, the conclusion is false by a
counterexample

f(x,y) = x*y*(x^2-y^2)/(x^2+y^2), f(0,0)=0.

However, if one of the two mixed partial derivatives is continuous, we
just need to establish that the other exists, and then they are equal.

This raises a question:
Has there been a study of sets of "non-Clairaut points", where f_xy and
f_yx exist and are different? Trivially, because of the above
counterexample, any finite set can be a non-Clairaut set. (Can you make
such a set bounded-and-infinite? Any restrictions? Etc.)

Cheers, ZVK(Slavek)
.

Relevant Pages

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