Lie derivation

The operator of Lie derivation is denoted by LD and its arguments are as follows: the first one contains the expression to be differentiated and the other ones are vector valued expressions according to which the differentiation happens.

LD(expr,X1,...,XN) $\Rightarrow$ $L$_X1&ldots;Xn  expr

Since Lie derivation can be performed not only with respect to vectors but also with respect to vector valued expressions, therefore their indices must not be omitted in the case of simple vectors either, as it is usual in a hand calculation:

The free index in the strictly vector valued arguments of the Lie derivatives can be arbitrary but the system performs index-changes automatically in order that they would coincide with the first abstract index according to the internal ordering:

Some rules are applied automatically in the case of Lie derivatives during the calculations (in the usual notation):

$L$_X  Yⁱ = - L_Y  Xⁱ, L_{X + Y}  T⁵i₁&ldots;i_r5j₁&ldots;j_s = L_X  Ti₁&ldots;i_rj₁&ldots;j_s + L_Y  Ti₁&ldots;i_rj₁&ldots;j_s, L_{a X}  T⁵i₁&ldots;i_r5j₁&ldots;j_s = a L_X  Ti₁&ldots;i_rj₁&ldots;j_s - ∑_k=1^r T⁵i₁..c.. i_r5j₁&ldots;j_s X⁵i_k a_;c + ∑_l=1^s T⁵i₁&ldots;i_r5j₁..c.. j_s X^c a₅;i_l,

where $Ti$₁&ldots;i_rj₁&ldots;j_s is a tensor of type (r,s), $X$, $Y$ are vector valued tensor expressions and $a$ is a constant or a scalar valued tensor expression. The Jacobi's identity can be activated by the command RELATION (see page ).