This paper presents the concept of the topological derivative in shape optimization, which is a mathematical tool used to analyze how a shape functional changes when a small hole is introduced into a domain. The topological derivative is defined as the limit of the difference in the shape functional when a small ball is removed from the domain, normalized by the volume of the ball. The paper provides a general definition of the topological derivative for arbitrary shape functionals and demonstrates its application to specific cases, such as elliptic equations and the elasticity system in two dimensions. The topological derivative is shown to be useful for solving shape optimization problems in structural mechanics. The paper also discusses the computation of the topological derivative for shape functionals that depend on solutions of elliptic equations. The method is constructive, meaning that the topological derivative can be evaluated for such functionals by solving the state equation and the appropriate adjoint state equation in the unperturbed domain. The paper provides examples of shape functionals, including those related to energy and compliance in linear elasticity. It shows that the topological derivative can be explicitly calculated for these functionals and provides formulas for the second derivative of the shape functional with respect to the size of the hole. The results are illustrated with numerical examples, including the Laplace equation and plane elasticity problems. The paper also discusses the application of the topological derivative to various types of domains, including those with cracks and reentrant corners, and provides conditions under which the results are applicable. The topological derivative is shown to be a powerful tool for shape optimization, as it can be used for a broad class of shape functionals and partial differential equations. The paper concludes with a detailed analysis of the topological derivative in the context of plane elasticity problems and provides formulas for the second derivative of the shape functional in this setting.This paper presents the concept of the topological derivative in shape optimization, which is a mathematical tool used to analyze how a shape functional changes when a small hole is introduced into a domain. The topological derivative is defined as the limit of the difference in the shape functional when a small ball is removed from the domain, normalized by the volume of the ball. The paper provides a general definition of the topological derivative for arbitrary shape functionals and demonstrates its application to specific cases, such as elliptic equations and the elasticity system in two dimensions. The topological derivative is shown to be useful for solving shape optimization problems in structural mechanics. The paper also discusses the computation of the topological derivative for shape functionals that depend on solutions of elliptic equations. The method is constructive, meaning that the topological derivative can be evaluated for such functionals by solving the state equation and the appropriate adjoint state equation in the unperturbed domain. The paper provides examples of shape functionals, including those related to energy and compliance in linear elasticity. It shows that the topological derivative can be explicitly calculated for these functionals and provides formulas for the second derivative of the shape functional with respect to the size of the hole. The results are illustrated with numerical examples, including the Laplace equation and plane elasticity problems. The paper also discusses the application of the topological derivative to various types of domains, including those with cracks and reentrant corners, and provides conditions under which the results are applicable. The topological derivative is shown to be a powerful tool for shape optimization, as it can be used for a broad class of shape functionals and partial differential equations. The paper concludes with a detailed analysis of the topological derivative in the context of plane elasticity problems and provides formulas for the second derivative of the shape functional in this setting.