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} } ) $:

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

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

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

See also Lie differentiation.