- #1
binbagsss
- 1,254
- 11
Hi,
I have done part a) by using the expression given for the lie derivative of a vector field and noting that if ##w## is a vector field then so is ##wf## and that was fine.
In order to do part b) I need to use the expression given in the question but looking at a component of the lie derivative of ##u ## w.r.t ##v## instead..And I'm unsure how to get this expression .
Homework Equations
The Attempt at a Solution
I've looked it up and see it is:
$$(\mathcal L_vu)^a=[v,u]^a=v^c\partial_cu^a-u^c\partial_c v^a = v^cu^a_{,c}-u^cv^a_{,c}$$
for the ##a##th component of ##\mathcal L_v w##
this is what I need to use right?
I am unsure how you get this expression for the ath component from the vector field expression we are given , working with the ##(0,0) ## rank tensors: ##V=V^u \partial_u ##. Should it be obvious via a index/dimension analysis or is it even more obvious than that?
I see that the expression is consistent with being covariant , if ##\partial_u \to \nabla_u ## then the expression holds and is covariant, so could almost trial my way to the correct expression however, but would like a better understanding.
(For example say if we wanted the lie derivative of ##a##component of ##w## wrt to ##v## my guess would be to do something like :
##V^uW^v\partial_v - W^v\partial_v V^u ##
loosing the derivative associated with the vector field V- which I know isn't what we seek anyway. But I'm confused because it looks like the dummy indices have been interchanged in the second term too, before loosing the derivative in that term)
many thanks