Lie-Brackets in Local Coordinates

Let \(X\) and \(Y\) be smooth vector fields, \(X=X^i\frac{\partial }{\partial x^i}\) and \(Y=Y^i\frac{\partial }{\partial x^i}\) their representation in local coordinates for some coordinates \((x^i)\). Then the Lie-bracket \([X,Y]\) may be expressed in local coordinates as follows

\begin{equation*} [X,Y]=\Bigl(X^i\frac{\partial Y^j}{\partial x^i}-Y^i\frac{\partial X^j}{\partial x^i}\Bigr)\frac{\partial }{\partial x^j} \end{equation*}

or shorter

\begin{equation*} [X,Y]=(XY^j-YX^j)\frac{\partial }{\partial x^j}. \end{equation*}