Lie bracket of vector fields on $\Bbb R^{n}$ 
Please show how to solve? I am stack with lie bracket. Thank you. 
 A: By definition, we have
$$[X,Y]=XY-YX.$$
Therefore, we have
$$\left[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\right]=
\left(-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}\right)\frac{\partial}{\partial x}
-\frac{\partial}{\partial x}\left(-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}\right)$$
$$=\left(-y\frac{\partial^2}{\partial x^2}+x\frac{\partial^2}{\partial y\partial x}\right)
-\left(-y\frac{\partial^2}{\partial x^2}\right)
-\left(\frac{\partial}{\partial y}+x\frac{\partial^2}{\partial y\partial x}\right)=-\frac{\partial}{\partial y}$$
because 
$$\frac{\partial}{\partial x}\left(-y\frac{\partial}{\partial x}\right)=
\frac{\partial}{\partial x}(-y)\frac{\partial}{\partial x}-y\frac{\partial^2}{\partial x^2}=-y\frac{\partial^2}{\partial x^2}$$
and 
$$\frac{\partial}{\partial x}\left(x\frac{\partial}{\partial y}\right)=
\frac{\partial}{\partial x}(x)\frac{\partial}{\partial y}+x\frac{\partial^2}{\partial x\partial y}=\frac{\partial}{\partial y}+x\frac{\partial^2}{\partial y\partial x}.$$
