The mathematics required to analyse higher dimensional curved spaces and space-times is developed in this chapter. General coordinate transformations, tangent spaces, vectors and tensors are described. Lie derivatives and covariant derivatives are motivated and defined. The concepts of parallel transport and a connection is introduced and the relation between the Levi-Civita connection and geodesics is elucidated. Christoffel symbols the Riemann tensor are defined as well as the Ricci tensor, the Ricci scalar and the Einstein tensor, and their algebraic and differential properties are described (though technical details of the derivationa of the Rimeann tensor are let to an appendix).

Lagrangian discrete models are the most ancient and still most effective models used in mechanics to predict the behavior of complex systems. Their structure is presented here, in a very classical way, which follows the classical presentation given by Levi-Civita. Their flexibility is highlighted, together with their capacity to be used in very efficient numerical codes. They must be regarded as a preferred tool also in the formulation of problems in the theory of metamaterials.