Lie bracket of vector fields ($[X,Y]$)

The Lie bracket is the vector field defined as a commutator of derivations,In coordinates, if $X=X^i{\partial }_i$ and $Y=Y^i{\partial }_i$, then $[X,Y{]}^k=X^i{\partial }_i{Y}^k-Y^i{\partial }_i{X}^k$.

For smooth vector fields $X,Y$, the local flows commute wherever both compositions are defined exactly when $[X,Y]=0$. The derivative of the pullback of $Y$ by the $X$ flow is the pullback of $[X,Y]$. A zero bracket therefore makes $Y$ invariant under that flow; uniqueness of solutions to its ordinary differential equation gives commutation. Conversely commutation implies invariance, whose derivative gives the zero bracket. Neither field need have a complete flow.

Under a change of coordinates, the second-derivative terms arising from the chain rule cancel because their coordinate indices are symmetric while the coefficient $X^i{Y}^j-Y^i{X}^j$ is antisymmetric. Consequently $[X,Y]$ transforms as a vector field.

Every pair of coordinate basis fields commutes:Therefore a frame containing two fields with nonzero Lie bracket of vector fields cannot be induced by a coordinate system.

On the punctured Euclidean plane, the orthonormal polar frame isIts bracket is $[\hat{r},\hat{\theta }]=-\hat{\theta }/r$, so it is not a coordinate basis even though $({\partial }_r,{\partial }_{\theta })$ is one.