This problem is of the book "An Introduction to pseudo differential operators" by Wong.
Find the symbol of each of the following partial differential operators on $\mathbb{R}^2.$
$\frac{\partial^2}{\partial {x_1}^2}+\frac{\partial^2}{\partial {x_2}^2}$
I have this:
The symbol of $\sum_{\alpha_1+\cdots +\alpha_n\leq m} a_{\alpha_1,\ldots, \alpha_n}(x){D_1}^{\alpha_1}\cdots {D_n}^{\alpha_n}$ with $D_{j}=-i\partial_j$ is $\sum_{\alpha_1+\cdots +\alpha_n\leq m} a_{\alpha_1,\ldots, \alpha_n}(x){\xi_1}^{\alpha_1}\cdots {\xi_n}^{\alpha_n}$
Then $\frac{\partial^2}{\partial {x_1}^2}+\frac{\partial^2}{\partial {x_2}^2}= \sum_{\alpha_1+\alpha_2\leq 2} a_{\alpha_1,\alpha_2}(x){D_1}^{\alpha_1}{D_2}^{\alpha_2}$ with $a_{0,0}=a_{0,1}=a_{1,0}=a_{1,1}=0,\ a_{2,0}=-1,\ a_{0,2}=-1$ because $a_{2,0}(x){D_1}^2+a_{0,2}{D_2}^{2}=-1(-i\partial_1)(-i\partial_1)+(-1(-i\partial_2)(-i\partial_2)=\frac{\partial^2}{\partial {x_1}^2}+\frac{\partial^2}{\partial {x_2}^2}$ then the symbol is $-{\xi_1}^2+-{\xi_2}^2$
It is right? since I'm not sure I understood correctly
Looks good. Since we have the definition $D = i\partial$, $D^2 = -\partial^2$, or $\partial^2 = -D^2$, the symbol of $\partial_1^2 + \partial_2^2$ is just $-\xi_1^2 -\xi_2^2$, as you noted.