  In this paper, we perform a rigourous version of shape and topological
derivatives for optimizations problems under constraint Helmoltz problems. A
shape and topological optimization problem is formulated by introducing cost
functional. We derive first by considering the lagradian method the shape
derivative of the functional. It is also proven a topological derivative with
the same approach. An application to several unconstrained shape functions
arising from differential geometry are also given.
