= Prolongation of a vector field
= Prolongations of a vector field
{synonym}
For $V=\xi(x,u)\partial_x+\eta(x,u)\partial_u$, its prolongation to derivatives through order $n$ is
$$
\operatorname{pr}^{(n)}V
=V+\sum_{j=1}^n\eta^{(j)}\partial_{u^{(j)}},
\qquad
\eta^{(j)}=D_x\eta^{(j-1)}-u^{(j)}D_x\xi.
$$
It generates a symmetry of $\Delta=0$ exactly when $\operatorname{pr}^{(n)}V(\Delta)$ vanishes on the equation manifold.
Prolongation lifts a <vector field> to the <jet bundle of maps> so that its flow acts on derivatives of the dependent variables as well as on their values.
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