The Lie bracket is the vector field defined as a commutator of derivations,In coordinates, if and , then .
Articles by others on the same topic
The Lie bracket of vector fields is an operation that takes two differentiable vector fields \( X \) and \( Y \) defined on a smooth manifold and produces another vector field, denoted \( [X, Y] \). This operation is essential in the study of the geometry of manifolds and plays a crucial role in various areas of differential geometry and mathematical physics.