The Lie derivative of tensor T_(a b) with respect to the vector field X is defined by ℒ_X T_(a b) congruent lim_(δ x->0) (T_(a b)^, (x') - T_(a b)(x))/(δ x). Explicitly, it is given by ℒ_X T_(a b) = T_(a d) X^d_(, b) + T_(d b) X^d_(, a) + T_(a b, e) X^e, where X_(, a) is a comma derivative. The Lie derivative of a metric tensor g_(a b) with respect to the vector field X is given by ℒ_X g_(a b) = X_(a;b) + X_(b;a) = 2X_(a;b), where X_(a, b) denotes the symmetric tensor part and X_(a;b) is a covariant derivative.
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