Lie bracket of vector fields on $R^2$

Question

Compute the Lie bracket$$\Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big]$$

on $R^2$

Can you help me please?

Answer 1

I am not 100% sure that this is what you want, but I think that $$\begin{align} \Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big] &= \left(-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}\right)\frac{\partial}{\partial x} - \frac{\partial}{\partial x}\left(-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}\right) \\ &= -y\frac{\partial^2}{\partial x^2}+x\frac{\partial}{\partial y}\frac{\partial}{\partial x} +y\frac{\partial^2}{\partial x^2} - \frac{\partial}{\partial x}\left(x\frac{\partial}{\partial y}\right) \\ &= \dots \end{align} $$

Use the product rule on the last part: $$\begin{align} \frac{\partial}{\partial x}\left(x\frac{\partial}{\partial y}\right) = \frac{\partial}{\partial y} + x \frac{\partial^2}{\partial x\partial y} \end{align} $$

Answer 2

[−y∂/∂x+x∂/∂y,∂/∂x]=(−y∂/∂x+x∂/∂y)(∂/∂x)−(∂/∂x)(−y∂/∂x+x∂/∂y)= - ∂/∂y