show that mixed partials are equal

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• Jan 11th 2012, 08:18 AM
mathrookie2012
show that mixed partials are equal
I need help to show that if f(x,y) is a polynomial then the mixed partials f(xy) and f(yx) are equal.
• Jan 11th 2012, 11:59 AM
MarceloFantini
Re: show that mixed partials are equal
Is the polynomial given? Is there more information? That's Schwarz's theorem for polynomials.
• Jan 11th 2012, 12:38 PM
chisigma
Re: show that mixed partials are equal
The Schwartz's theorem extablishes that, given an f(x.y), if in some $(x_{0},y_{0})$ the partial derivatives $f^{''}_{xy}$ and $f^{''}_{yx}$ both exist and are continuos, then $f^{''}_{xy} (x_{0},y_{0})= f^{''}_{yx} (x_{0},y_{0})$. If f(x,y) is a polynomial, then the condition is satisfied...

Kind regards

$\chi$ $\sigma$
• Jan 11th 2012, 01:50 PM
mathrookie2012
Re: show that mixed partials are equal
the polynomial is not given and there is no more information given. I have tried to find a proof of it but I cannot seem to find one.
• Jan 11th 2012, 07:40 PM
chisigma
Re: show that mixed partials are equal
By definition a polynomial in the variables x abd y is a function that contains only sums, multiplications and powers of the variables. This type of function has partial derivatives of all order continuos, so that the Schwartz's theorem [see my previous post...] can be applied...

Kind regards

$\chi$ $\sigma$
• Jan 11th 2012, 08:26 PM
FernandoRevilla
Re: show that mixed partials are equal
Quote:

Originally Posted by mathrookie2012
the polynomial is not given and there is no more information given. I have tried to find a proof of it but I cannot seem to find one.

One way (without using Schwartz's Lemma): every polynomial $f(x,y)\in\mathbb{R}[x,y]$ can be written in the way

$f(x,y)=\sum_{i\geq 0,\;j\geq 0}a_{ij}x^iy^j$

where $i,j$ are integers and the coefficients $a_{ij}\in\mathbb{R}$ are finitely many non zero. Then,

$\frac{{\partial f}}{{\partial x}}=\sum_{i\geq 1,\;j\geq 0}ia_{ij}x^{i-1}y^j\;,\quad \frac{{\partial^2 f}}{{\partial x\partial y}}=\sum_{i\geq 1,\;j\geq 1}ija_{ij}{x^{i-1}y^{j-1}$

Now, find $\frac{{\partial^2 f}}{{\partial y\partial x}}$ and compare with $\frac{{\partial^2 f}}{{\partial x\partial y}}$ .
• Jan 12th 2012, 04:38 AM
mathrookie2012
Re: show that mixed partials are equal
To be able to use the Schwartz theorem the partial derivative has to exist. How do I sshow they exist?
• Jan 12th 2012, 04:50 AM
FernandoRevilla
Re: show that mixed partials are equal
Quote:

Originally Posted by mathrookie2012
To be able to use the Schwartz theorem the partial derivative has to exist. How do I sshow they exist?

With this problem, I'm afraid we'll have to reproduce the History of Mathematics. :)
• Jan 12th 2012, 08:12 AM
mathrookie2012
Re: show that mixed partials are equal
so has far as showing that the partials are equal, what is the best method to take?
• Jan 12th 2012, 08:16 AM
mathrookie2012
Re: show that mixed partials are equal
Show that if f(x; y) is a polynomial of x; y 2 R, then the mixed partials fxy and fyx
are equal.

This is how the problem is written
• Jan 12th 2012, 10:10 AM
FernandoRevilla
Re: show that mixed partials are equal
Quote:

Originally Posted by mathrookie2012
Show that if f(x; y) is a polynomial of x; y 2 R, then the mixed partials fxy and fyx
are equal. This is how the problem is written

That is clear, but the problem is that we don't know exactly what properties are allowed to use. Depends on your syllabus. At any case we have provided you two correct, different and (I guess sufficient) methods.
• Jan 12th 2012, 11:55 AM
mathrookie2012
Re: show that mixed partials are equal
Thanks so much for the help guys
• Jan 16th 2012, 04:59 AM
mathrookie2012
Re: show that mixed partials are equal
How would you find second partial of y
• Jan 16th 2012, 05:06 AM
FernandoRevilla
Re: show that mixed partials are equal
Quote:

Originally Posted by mathrookie2012
How would you find second partial of y