Lie derivative Definition and 59 Threads

Can somone remind me how to see that the Lie derivative of a vector field, defined as (L_XY)_p=\lim_{t\rightarrow 0}\frac{\phi_{-t}_*Y_{\phi_t(p)}-Y_p}{t} is actually equal to [X,Y]_p?

I'm trying to use an example to make sense out of the equation \mathcal{L}_X = d\circ i_X + i_X\circ d. Some simple equations: \omega = \omega^1 dx_1 + \omega^2 dx_2 i_X\omega = X_1\omega^1 + X_2\omega^2 (d\omega)^{11} = (d\omega)^{22} = 0,\quad (d\omega)^{12} =...

Suppose you have a spacetime with an observer at rest at the origin, and the surface at t = 0 going through the origin, and passing through the surface there are geodesics along increasing time. Then as you get a small ways away from the surface, the geodesics start to deviate from each other...

Suppose we define the Lie derivative on a tensor T at a point p in a manifold by \mathcal{L}_V (T) = \lim_{\epsilon \to 0}\frac{\varphi_{-\epsilon \ast}T(\varphi_\epsilon(p))- T(p)}{\epsilon} where V is the vector field which generates the family of diffeomorphisms \varphi_t. If T is just an...

what is the physical significance of the lie derivative? What is its purpose in tensor analysis?

i was curious as to what exactly this is and more importantly, what motivates it. what are its applications?

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

Let M be a diff. manifold, X a complete vectorfield on M generating the 1-parameter group of diffeomorphisms \phi_t. If I now define the Lie Derivative of a real-valued function f on M by \mathscr{L}_Xf=\lim_{t\rightarrow 0}\left(\frac{\phi_t^*f-f}{t}\right)=\frac{d}{dt}\phi_{t}^{*}f |_{t=0}...

I'd like an example of calculating the Lie derivative of a one-form with respect to a vector field, for example, the one-form \omega = 3 dx_1 + 4x dx_2 with the vector field X = 7x \frac{\partial }{\partial x_1} + 2 \frac{\partial }{\partial x_2} Any input would be...