This module focuses on mathematical fundamentals - especially tensor calculation. These enable the mathematical formulation of central mechanical quantities by means of tensors of different levels. In detail, the tensor algebra and aspects of tensor analysis as well as the integral theorems that can be formulated with it are treated, which are required, for example, for the closed formulation of material models and thermodynamic balance equations. Elementary vector properties, operations, and transformations are covered at the beginning of the module and then extended and applied to second-level tensors. This includes additive, multiplicative, and spectral decompositions, as well as the Cayley-Hamilton theorem. By analogy, fourth level tensors are introduced and their representation in Voigt and Kelvin notation, among others, is treated. In the subsequent treatment of tensor analysis, basic topics such as directional derivatives and elementary quantities such as the gradient, divergence, and rotation operators are introduced and discussed. Finally, these considerations are extended to general and nonorthonormal basis systems. In the exercises the focus is on the independent implementation/programming of the contents discussed in the lecture.