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Math Help - show that mixed partials are equal

  1. #1
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    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.
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    Newbie MarceloFantini's Avatar
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    Re: show that mixed partials are equal

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

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    \chi \sigma
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    MHF Contributor FernandoRevilla's Avatar
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    Re: show that mixed partials are equal

    Quote Originally Posted by mathrookie2012 View Post
    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}} .
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    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?
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    MHF Contributor FernandoRevilla's Avatar
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    Re: show that mixed partials are equal

    Quote Originally Posted by mathrookie2012 View Post
    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.
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    Re: show that mixed partials are equal

    so has far as showing that the partials are equal, what is the best method to take?
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    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
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  11. #11
    MHF Contributor FernandoRevilla's Avatar
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    Re: show that mixed partials are equal

    Quote Originally Posted by mathrookie2012 View Post
    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.
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    Re: show that mixed partials are equal

    Thanks so much for the help guys
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  13. #13
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    Re: show that mixed partials are equal

    How would you find second partial of y
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  14. #14
    MHF Contributor FernandoRevilla's Avatar
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    Re: show that mixed partials are equal

    Quote Originally Posted by mathrookie2012 View Post
    How would you find second partial of y
    Your question has no sense. Please, reword it.
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  15. #15
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    Re: show that mixed partials are equal

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