I need help to show that if f(x,y) is a polynomial then the mixed partials f(xy) and f(yx) 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
$\displaystyle \chi$ $\displaystyle \sigma$
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
$\displaystyle \chi$ $\displaystyle \sigma$
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 $\displaystyle \frac{{\partial^2 f}}{{\partial x\partial y}}$ .