Here is an approach for lie derivative. And i would like to know how wrong is it. Assuming lie derivative of a vector field measures change of a vector field along a vector field, take a coordinate system, xi , and the vector field fi along which Ti is being changed. I go this way, i take the vector 'T' at point xi, transform the coordinate as- xi -> xi + efi= x'i, e nearly zero. Now Ti(x) is transformed to T'i(x'). Now I subtract T'(x') with T(x). We would do the same thing to find directional derivative of a scalar function- S( x + ef)- S(x)= S(x')-S(x) Now back to vector T. T'j(x')= Ti(x)dx'j/dxi. Now take Ti(x') = Ti(x) + efk[tex]\nabla[/tex]kTi. Since T(x) is a vector field. But I don't ask for a definition of connection. We use arbitrary connection and try to see if T(x') is completely specified by T(x) by requiring specification of connection [tex]\nabla[/tex]. So T'j(x')= Ti(x') - e( fk[tex]\nabla[/tex]kTi(x) - Tk(x')dfi /dxk To first approximation. This finally gives, considering first order in e, T'j(x')= Ti(x) - e( fk[tex]\nabla[/tex]kTi(x) - Tk(x')[tex]\nabla[/tex]kfi) = Ti(x) + e(lie derivative required) + e2(terms..) As is seen here, the final form does not include the notion of a definite vector field for T. Even connection [tex]\nabla[/tex] can be removed from lie derivative for vector. I need to know if it is correct, the assumption that lie derivative can be considered as change in tensor, not necessarily a tensor field. Or is a precisely defined tensor field property already used here?