The paper by J. Sokolowski and A. Zochowski introduces the concept of topological derivatives for shape functionals, providing a method to solve shape optimization problems in structural mechanics. The topological derivative is defined as the limit of the ratio of the change in the functional to the size of the perturbation as the perturbation tends to zero. The authors provide examples for elliptic equations and the elasticity system in the plane, demonstrating how the topological derivative can be used to evaluate shape functionals for solutions of partial differential equations. The paper includes detailed mathematical derivations and numerical examples to illustrate the application of the topological derivative in various scenarios, such as Laplace equation and plane elasticity problems. The results are applicable to a broad class of shape functionals and partial differential equations, making the topological derivative a valuable tool in shape optimization.The paper by J. Sokolowski and A. Zochowski introduces the concept of topological derivatives for shape functionals, providing a method to solve shape optimization problems in structural mechanics. The topological derivative is defined as the limit of the ratio of the change in the functional to the size of the perturbation as the perturbation tends to zero. The authors provide examples for elliptic equations and the elasticity system in the plane, demonstrating how the topological derivative can be used to evaluate shape functionals for solutions of partial differential equations. The paper includes detailed mathematical derivations and numerical examples to illustrate the application of the topological derivative in various scenarios, such as Laplace equation and plane elasticity problems. The results are applicable to a broad class of shape functionals and partial differential equations, making the topological derivative a valuable tool in shape optimization.