The Mixed Derivative Theorem and the Ring Structure of Differential Operators

Mixed derivative theorem

Let $E$$E$ is an open subset of ${\mathbb{R}}^n$$\mathbb R^n$, $f:E\to \mathbb{R}\in C^2$$f:E\to\mathbb R\in C^2$ , $\mathrm{\forall }1\le i,j\le n$$\forall 1\le i,j\le n$ and $\mathrm{\forall }x\in E$$\forall x\in E$, ${\partial }_i{\partial }_j(f)|x={\partial }_j{\partial }_i(f)|x$$\part_i\partial_j(f)|x=\part_j\partial_i(f)|x$

$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}.$$\rm Proof.$

Let $a={\partial }_i{\partial }_j(f){|}_{x_0},a^{\prime }={\partial }_j{\partial }_i(f{)}_{x_0}$$a=\partial_i\partial_j(f)|_{x_0},a'=\partial_j\partial_i (f)_{x_0}$

We need to prove that $a=a^{\prime }$$a=a'$

Because of $f\in C^2$$f\in C^2$，$\mathrm{\forall }ϵ>0,\mathrm{\exists }\delta$$\forall\epsilon> 0,\exists \delta$, when $|x-x_0|\le \delta |,|{\partial }_i{\partial }_j(f){|}_{x_0}-a|\le ϵ$$|x-x_0|\le\delta|,|\partial_i\partial_j(f)|_{x_0}-a|\le\epsilon$

Similarly, $\mathrm{\forall }ϵ>0,\mathrm{\exists }\delta$$\forall\epsilon> 0,\exists \delta$, when $|x-x_0|\le \delta |,|{\partial }_j{\partial }_i(f){|}_{x_0}-a^{\prime }|\le ϵ$$|x-x_0|\le\delta|,|\partial_j\partial_i(f)|_{x_0}-a'|\le\epsilon$

Now, consider $F:=f(\delta e_i+\delta e_j+x_0)-f(\delta e_j+x_0)-(f(\delta e_i+x_0)-f(x_0))$$F:= f(\delta e_i+\delta e_j+x_0)-f(\delta e_j+x_0)-(f(\delta e_i+x_0)-f(x_0))$

According to FTC, $\begin{array}{c}F={\int }_0^{\delta }{\partial }_{x_i}f(x_0+\delta e_j+x_i{e}_i)-{\partial }_{x_i}f(x_0+x_i{e}_i)d{x}_i\end{array}$$F=\int_0^\delta \partial _{x_i}f(x_0+\delta e_j+x_i e_i) - \partial _{x_i}f(x_0+x_i e_i)dx_i\\$

According to the Mean value theorem, $\mathrm{\forall }{x}_i,\mathrm{\exists }0\le x_j\le \delta ,{\partial }_{x_i}f(x_0+\delta e_j+x_i{e}_i)-{\partial }_{x_i}f(x_0+x_i{e}_i)=\delta {\partial }_{x_j}{\partial }_{x_i}f(x_0+x_i{e}_i+x_j{e}_j)$$\forall x_i,\exists 0\le x_j\le\delta,\partial _{x_i}f(x_0+\delta e_j+x_i e_i) - \partial _{x_i}f(x_0+x_i e_i)=\delta \partial_{x_j}\partial_{x_i}f(x_0+x_i e_i+x_j e_j)$

Observe that $x_i,x_j\le \delta$$x_i,x_j\le\delta$

Thus $|{\partial }_{x_i}f(x_0+\delta e_j+x_i{e}_i)-{\partial }_{x_i}f(x_0+x_i{e}_i)-\delta a|\le \delta ϵ$$|\partial _{x_i}f(x_0+\delta e_j+x_i e_i) - \partial _{x_i}f(x_0+x_i e_i)-\delta a|\le\delta\epsilon$

Therefore $\begin{array}{c}{\int }_0^{\delta }|{\partial }_{x_i}f(x_0+\delta e_j+x_i{e}_i)-{\partial }_{x_i}f(x_0+x_i{e}_i)-\delta a|d_{x_i}=|F-{\delta }^{2}a|\le {\delta }^2ϵ\end{array}$$\int_0^\delta |\partial _{x_i}f(x_0+\delta e_j+x_i e_i) - \partial _{x_i}f(x_0+x_i e_i)-\delta a|d_{x_i}=|F-\delta^2 a|\le\delta^2\epsilon\\$

And change the order of $i,j$$i,j$, we have $|F-{\delta }^2{a}^{\prime }|\le {\delta }^2ϵ$$|F-\delta^2 a'|\le\delta^2\epsilon$

According to triangle inequality, $|{\delta }^{2}a-F+F-{\delta }^2{a}^{\prime }|\le |F-{\delta }^{2}a|+|F-{\delta }^2{a}^{\prime }|\le 2{\delta }^2ϵ$$|\delta^2 a- F+F-\delta^2 a'|\le |F-\delta ^2a|+|F-\delta^2 a'|\le 2\delta^2\epsilon$

Thus $|a-a^{\prime }|\le 2ϵ$$|a-a'|\le 2\epsilon$

Then we can generalize it to $f:E\to {\mathbb{R}}^m$$f:E\to \mathbb R^m$

And consider $C^{\mathrm{\infty }}:E\subseteq {\mathbb{R}}^n\to \mathbb{R}$$C^{\infty}:E\subseteq \mathbb R^n\to\mathbb R$

Observe that $\mathbb{R}[{\partial }_1,{\partial }_2,...,{\partial }_n]\cong \mathbb{R}[x_1,x_2,...,x_n]$$\mathbb R[\part_1,\part_2,...,\part_n]\cong\mathbb R[x_1,x_2,...,x_n]$