Pullback of a mixed tensor by a diffeomorphism ($({\varphi }^{\ast }T{)}_p=[(d{\varphi }_p{)}^{-1}{]}^{\otimes r}\otimes [(d{\varphi }_p{)}^{\ast }{]}^{\otimes s}{T}_{\varphi (p)}$)

For a diffeomorphism, pull a type $(r,s)$ tensor field back by the inverse differential on vector slots and the dual differential on covector slots. Invertibility distinguishes this construction from pullback of purely covariant tensors along an arbitrary smooth map. This operation preserves tensor products and tensor contractions, supplying the flow definition of the Lie derivative of a tensor field.