show that mixed partials are equal

**mathrookie2012** · January 11th 2012, 07:18 AM

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

**MarceloFantini** · January 11th 2012, 10:59 AM

Is the polynomial given? Is there more information? That's Schwarz's theorem for polynomials.

**chisigma** · January 11th 2012, 11:38 AM

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

$\chi$ $\sigma$

**mathrookie2012** · January 11th 2012, 12:50 PM

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.

**chisigma** · January 11th 2012, 06:40 PM

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

$\chi$ $\sigma$

**FernandoRevilla** · January 11th 2012, 07:26 PM

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 $\frac{{\partial^2 f}}{{\partial x\partial y}}$ .

**mathrookie2012** · January 12th 2012, 03:38 AM

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

**FernandoRevilla** · January 12th 2012, 03:50 AM

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.

**mathrookie2012** · January 12th 2012, 07:12 AM

so has far as showing that the partials are equal, what is the best method to take?

**mathrookie2012** · January 12th 2012, 07:16 AM

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

**FernandoRevilla** · January 12th 2012, 09:10 AM

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.

**mathrookie2012** · January 12th 2012, 10:55 AM

Thanks so much for the help guys

**mathrookie2012** · January 16th 2012, 03:59 AM

How would you find second partial of y

**FernandoRevilla** · January 16th 2012, 04:06 AM

Originally Posted by mathrookie2012

How would you find second partial of y

Your question has no sense. Please, reword it.

**mathrookie2012** · January 16th 2012, 04:45 AM

You told me to find d^2f/dydx. But how?

Thread: show that mixed partials are equal

LinkBack

Thread Tools

Display

show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Re: show that mixed partials are equal

Similar Math Help Forum Discussions

[SOLVED] Not C^2, but mixed partials exist and are equal.

Another "mixed partials" question

"Mixed partials" question

Inequality of mixed partials

Help me please.... Mixed Partials

Search Tags