Lie bracket of vector fields

Summary

The Lie bracket of two vector fields $X$ and $Y$ on a smooth manifold is a third vector field $[X, Y]$. It measures how much the flows of $X$ and $Y$ fail to commute. It is zero exactly when, for short times, moving along $X$ then $Y$ gives the same result as moving along $Y$ then $X$. With this bracket the vector fields on a manifold form a Lie algebra. It appears in the Frobenius theorem, in the definition of curvature and torsion, and in control theory.