 # Basics of Mathematics (Winter-Term 2023)

 Lecturer: Prof. Dr. Thomas Stiehl Tutor: Chenxu Zhu Dates: Lecture: Mo, 12:45-14:15 (different venues, see RWTH-online) Exercise: Fr, 12:30-14:00 (different venues, see RWTH-online) Start: 9.10.2023 Lecture Notes: Lecture notes are updated after the lectures. Exercise sheets to be completed as homework: The written solutions have to be handed to the tutor before the tutorium starts. Exercise sheets to be considered in the tutorium: These exercise sheets do not have to be handed in! Summary: The course recapitulates relevant topics of engineering mathematics. It begins with a summary of fundamental concepts such as mathematical proofs, sets and numbers. Relevant topics of linear algebra including vector spaces, linear transformations, matrix calculus, determinants and eigenvalues are covered. Important foundations of calculus such as limits, sequences and series are summarized in a compressed form. A major part of the course deals with multidimensional differential and integral calculus including extreme values, fundamental theorem of calculus, Fourier analysis, Divergence Theorem, Stokes’ Theorem and Green’s Theorem. A primer on ordinary differential equations covering linear systems, fixed points and linearized stability analysis is also given. The course is designed as a refresher with a focus on applications and practical calculations. It does not provide a comprehensive derivation of the underlying mathematical theories. Topics: Techniques of mathematical proofs (direct proof, indirect proof), complete induction Elementary set theory Numbers (natural numbers, integers, rational numbers, real numbers, complex numbers) Functions and basic properties of functions (injectivity, surjectivity, bijectivity, linearity) Polynomials and the fundamental theorem of algebra Linear algebra (vector spaces, matrix calculus, linear transformations, eigenvectors, eigenvalues, determinants, scalar products, optional: Jordan normal form, principal axis theorem) Norms and metrics Limits and continuity Sequences and series Differential calculus (Differentiation rules, extreme values, Jacobian, Hessian; optional: implicit function theorem, inverse function theorem) Important functions and their approximation (spaces of continuous and differentiable functions, power series, Taylor series) Integral calculus (multi-dimensional integration, fundamental theorem of calculus, change of variables, Divergence Theorem, Stokes‘ Theorem, Green‘s Theorem, Fourier transform; optional: convolutions, mollification) Ordinary differential equations (existence and uniqueness of solutions, linear initial value problems, fixed points, linearized stability; optional: basic bifurcation theory) Basics of statistical testing Optional: important partial differential equations (diffusion equation, transport equation, wave equation)