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

Printable View

- January 11th 2012, 08:18 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.

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

- January 11th 2012, 12:38 PMchisigmaRe: show that mixed partials are equal
The Schwartz's theorem extablishes that, given an f(x.y), if in some the partial derivatives and both exist and are continuos, then . If f(x,y) is a polynomial, then the condition is satisfied...

Kind regards

- January 11th 2012, 01:50 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.

- January 11th 2012, 07:40 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

- January 11th 2012, 08:26 PMFernandoRevillaRe: show that mixed partials are equal
- January 12th 2012, 04:38 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?

- January 12th 2012, 04:50 AMFernandoRevillaRe: show that mixed partials are equal
- January 12th 2012, 08:12 AMmathrookie2012Re: show that mixed partials are equal
so has far as showing that the partials are equal, what is the best method to take?

- January 12th 2012, 08:16 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 - January 12th 2012, 10:10 AMFernandoRevillaRe: show that mixed partials are equal
- January 12th 2012, 11:55 AMmathrookie2012Re: show that mixed partials are equal
Thanks so much for the help guys

- January 16th 2012, 04:59 AMmathrookie2012Re: show that mixed partials are equal
How would you find second partial of y

- January 16th 2012, 05:06 AMFernandoRevillaRe: show that mixed partials are equal
- January 16th 2012, 05:45 AMmathrookie2012Re: show that mixed partials are equal
You told me to find d^2f/dydx. But how?