# Intuitively what's the difference between Lie Derivative and Covariant Derivative?

sit.think.solve
Loosely speaking or Intuitively how should one understand the difference between Lie Derivative and Covariant derivative? Definitions for both sounds awfully similar...

Staff Emeritus
Just to spotlight one of these: The Lie derivative L_X(Y) is basically a dynamical invariant. It measures how compatible the flows of the vector fields are, i.e. how much they commute with each other. If X is generated by the flow $\phi_t$ and Y is generated by $\psi_s$, then the Lie derivative [X,Y] at point P is the tangent vector at time 0 of the curve given by: $t \mapsto \psi_{-t} \circ \phi_{-t} \circ \psi_t \circ \phi_t (P)$.
In particular, if $\phi_s \circ \psi_t=\psi_t \circ \phi_s$ for all s and t, then [X,Y]=0.