## Lie Bracket Problems

If $X$, $Y$ are vector fields on a manifold $M$ and $f$, $g:{M}\rightarrow {R}$ differentiable functions, show that
$[fX,gY]=fg[X,Y]+f(Xg)Y-g(Yf)X$

Also if $X=x_{2}\frac{\partial}{\partial{x_{1}}}-x_{2}\frac{\partial}{\partial{x_{2}}}$, $Y=x_{3}\frac{\partial}{\partial{x_{2}}}-x_{2}\frac{\partial}{\partial{x_{3}}}$, $Z=\frac{\partial}{\partial{x_{1}}}+\frac{\partial} {\partial{x_{2}}}+\frac{\partial}{\partial{x_{3}}}$ are vector fields on $R^3$, calculate the components of $[X,Y]$, $[X,Z]$, $[Y,Z]$.

Finally, let $(U,f)$ be a map of the $m$-dimensional manifold $M$ and $X$, $Y$ be two dimensional fields onto $U$. If $\frac{\partial}{\partial{x_{i}}} (i=1,...,m)$ are the basic vector fields of $(U,f)$ and $X=\sum_{i=1}^{m}a_{i}\frac{\partial}{\partial{x_{i }}}$, $Y=\sum_{i=1}^{m}b_{i}\frac{\partial}{\partial{x_{i }}}$ where $a_{i}$, $b_{i}
\in C^{oo}(U,R)$
show that $[X,Y]=\sum_{i=1}^{m}(\sum_{j=1}^{m}(a_{j}\frac{\partial {b_{i}}}{\partial{x_{j}}}-b_{j}\frac{\partial{a_{i}}}{\partial{x_{j}}})\frac {\partial}{\partial{x_{i}}}$.

( $[,]$ denotes the Lie bracket on vector fields) I am not familiar with the Lie bracket at all, so I would be happy if someone helped!