  The treatment of higher order perturbations of branes is considered using a
covariant variational approach. This covariant variational approach brings to
the forefront the geometric structure of the underlying perturbation theory, as
opposed to a more commonly used `direct approach', that ignores the variational
origin. In addition, it offers a clear calculational advantage with respect to
so called `gauge fixed' treatments that distinguish tangential and normal
modes, as it emphasizes the symmetries of the geometric models that describe
the brane dynamics. We restrict our attention to a brane action that depends at
most on first derivatives of the embedding functions of the worldvolume spanned
by the brane in its evolution. We consider first and second variations of the
action that describes the brane dynamics. The first variation produces the
equations of motion, as is well known. In the second variation we derive the
Jacobi equations for these kind of models, and we emphasize the role of the
Hessian matrix. This is extended to third order in variations, first in a flat
and then in a curved spacetime background. Further, we specialize to the
relevant case of the Dirac-Nambu-Goto action that describes extremal branes.
The proper setting of a covariant variational approach allows to go in
principle to geometric models that depend on higher derivatives of the
embedding functions, and higher order perturbations, with the due complications
involved, but with a solid framework in place.
