The prolongation formula for tensor fields
Résumé
We derive a generalization of the prolongation formula for
tensor-valued functions, by taking into account the natural action
of diffeomorphisms on tensor fields. This differs from the usual
procedure, where such fields are viewed as defining
``multi-component scalars;'' it replaces the ordinary derivative
in the expression of the ``characteristic'' by a Lie derivative.
The resulting version of Noether's theorem takes a form closer to
what is familiar to theoretical physicists. A very simple proof of
the conformal invariance of the Maxwell Lagrangian also follows
from this procedure. The Eshelby tensor is also readily obtained,
as a further illustration of the practical use of this new
prolongation formula.