# Lie derivative

of a tensor field $Q$ in the direction of a vector field $X$ on a manifold $M$

The tensor field ${\mathcal L} _ {X} Q$ on $M$, of the same type as $Q$, given by the formula

$$( {\mathcal L} _ {X} Q ) _ {x} = \lim\limits _ {t \rightarrow 0 } \frac{1}{t} (( \phi _ {t} ^ {*} Q ) _ {x} - Q _ {x} ) ,\ x \in M ,$$

where $\phi _ {t} ^ {*}$ is the local one-parameter group of transformations of the space of tensor fields generated by the vector field $X$. In local coordinates $x ^ {i}$, the Lie derivative of a tensor field $Q = ( Q _ {j _ {1} \dots j _ {l} } ^ {i _ {1} {} \dots i _ {k} } )$ of type $( k , l )$ in the direction of the vector field $X = ( X ^ {i} )$ has coordinates $( \partial _ {i} = \partial / {\partial x ^ {i} } )$:

$$( {\mathcal L} _ {X} Q ) _ {j _ {1} \dots j _ {l} } ^ {i _ {1} \dots i _ {k} } = X ^ {i} \partial _ {i} Q _ {j _ {1} \dots j _ {l} } ^ {i _ {1} \dots i _ {k} } +$$

$$- \sum _ {\alpha = 1 } ^ { k } \partial _ {i} X ^ {i _ \alpha } Q _ {j _ {1} \dots j _ {k} } ^ {i _ {1} \dots \widehat{i} _ \alpha ii _ {\alpha + 1 } \dots i _ {k} } +$$

$$+ \sum _ {\beta = 1 } ^ { l } \partial _ {j _ \beta } X ^ {j} Q _ { j _ {1} \dots \widehat{j} _ \beta jj _ {\beta + 1 } \dots j _ {l} } ^ {i _ {1} \dots i _ {k} } .$$