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}<br />
\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!