Lie bracket of vector fields on $\Bbb R^{n}$

Question

Please show how to solve? I am stack with lie bracket. Thank you.

Answer 1

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 $$\frac{\partial}{\partial x}\left(x\frac{\partial}{\partial y}\right)= \frac{\partial}{\partial x}(x)\frac{\partial}{\partial y}+x\frac{\partial^2}{\partial x\partial y}=\frac{\partial}{\partial y}+x\frac{\partial^2}{\partial y\partial x}.$$