Consider two smooth vector fields \(X\) and \(Y\). The Lie-bracket combines both and provides a new smooth vector field. It is defined as follows:
\begin{equation*} [X,Y]f:=XYf-YXf. \end{equation*}
Remarks
- One could think that \(XY\) is an easier choice. However, \(XY\) does not suffices the product rule in general, therefore it is not a tangent vector.