Abstract
The object of this paper is to further investigate the notion of shape and topological derivatives in the light of the general notion of Hadamard semidifferential for a function defined on a subset of a topological vector space.
The use of semitrajectories and the characterization of the adjacent tangent cone provide simple tools for defining Hadamard semi-differentials and differentials without a priori introduction of geometric structures such as, for instance, a differential manifold. Such a simple notion retains all the operations of the classical differential calculus, including the chain rule, for a large class of nondifferentiable functions, in particular, the norms and the convex functions. It also provides a direct access to functions defined on a lousy set or a manifold with boundary.
This direct approach is first illustrated in the context of the classical matrix subgroups of the general linear group GL(n) of invertible n×n matrices, which are the prototypes of Lie groups. For the shape derivative we have groups of diffeomorphisms of the Euclidean space ℝn with the composition operation, and the adjacent tangent cone is a linear space; for the topological derivative we have the group of characteristic functions with the symmetric difference operation and the adjacent tangent cone is only a cone at some points.