  We introduce new results about the shape derivatives of scalar- and
vector-valued functions, extending the results from (Dogan-Nochetto 2012) to
more general surface energies. They consider surface energies defined as
integrals over surfaces of functions that can depend on the position, the unit
normal and the mean curvature of the surface. In this work we present a
systematic way to derive formulas for the shape derivative of more general
geometric quantities, including the Gauss curvature (a new result not available
in the literature) and other geometric invariants (eigenvalues of the second
fundamental form). This is done for hyper-surfaces in the Euclidean space of
any finite dimension. As an application of the results, with relevance for
numerical methods in applied problems, we introduce a new scheme of Newton-type
to approximate a minimizer of a shape functional. It is a mathematically sound
generalization of the method presented in (Chicco-Ruiz et. al. 2016). We
finally find the particular formulas for the first and second order shape
derivatives of the area and the Willmore functional, which are necessary for
the Newton-type method mentioned above.

Optimization and Control;Numerical Analysis