I need help to show that if f(x,y) is a polynomial then the mixed partials f(xy) and f(yx) are equal.

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- Jan 11th 2012, 07:18 AMmathrookie2012show 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, 10:59 AMMarceloFantiniRe: show that mixed partials are equal
Is the polynomial given? Is there more information? That's Schwarz's theorem for polynomials.

- Jan 11th 2012, 11:38 AMchisigmaRe: show that mixed partials are equal
The Schwartz's theorem extablishes that, given an f(x.y), if in some $\displaystyle (x_{0},y_{0})$ the partial derivatives $\displaystyle f^{''}_{xy}$ and $\displaystyle f^{''}_{yx}$ both exist and are continuos, then $\displaystyle f^{''}_{xy} (x_{0},y_{0})= f^{''}_{yx} (x_{0},y_{0})$. If f(x,y) is a polynomial, then the condition is satisfied...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$ - Jan 11th 2012, 12:50 PMmathrookie2012Re: 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, 06:40 PMchisigmaRe: 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

$\displaystyle \chi$ $\displaystyle \sigma$ - Jan 11th 2012, 07:26 PMFernandoRevillaRe: show that mixed partials are equal
One way (without using Schwartz's Lemma): every polynomial $\displaystyle f(x,y)\in\mathbb{R}[x,y]$ can be written in the way

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

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

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

Now, find $\displaystyle \frac{{\partial^2 f}}{{\partial y\partial x}}$ and compare with $\displaystyle \frac{{\partial^2 f}}{{\partial x\partial y}}$ . - Jan 12th 2012, 03:38 AMmathrookie2012Re: 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, 03:50 AMFernandoRevillaRe: show that mixed partials are equal
- Jan 12th 2012, 07:12 AMmathrookie2012Re: 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, 07:16 AMmathrookie2012Re: 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, 09:10 AMFernandoRevillaRe: show that mixed partials are equal
- Jan 12th 2012, 10:55 AMmathrookie2012Re: show that mixed partials are equal
Thanks so much for the help guys

- Jan 16th 2012, 03:59 AMmathrookie2012Re: show that mixed partials are equal
How would you find second partial of y

- Jan 16th 2012, 04:06 AMFernandoRevillaRe: show that mixed partials are equal
- Jan 16th 2012, 04:45 AMmathrookie2012Re: show that mixed partials are equal
You told me to find d^2f/dydx. But how?