- #1
kent davidge
- 933
- 56
I subtracted the ##\mu##-th component of the Lie Derivative of a Vector ##U## along a vector ##V## from the ##\mu##-th component of the Covariant derivative of the same vector ##U## along the same vector ##V## and I got ##(\nabla_V U)^\mu - (\mathcal{L}_V U)^\mu = U^\nu \partial_\nu V^\mu - V^\nu U^\sigma \Gamma^\mu{}_{\nu \sigma}##
I know I should really say vector field in the above instead of vector. My question is if it's legitimate to perform such subtraction. If so, One notices that the two derivatives are the same when the basis and the vector field ##V## are constant.
I know I should really say vector field in the above instead of vector. My question is if it's legitimate to perform such subtraction. If so, One notices that the two derivatives are the same when the basis and the vector field ##V## are constant.