401-0251-00L Mathematics I
|Semester||Autumn Semester 2016|
|Lecturers||A. Cannas da Silva|
|Periodicity||yearly recurring course|
|Language of instruction||German|
|Abstract||This course covers mathematical concepts and techniques necessary to model, solve and discuss scientific problems - notably through ordinary differential equations.|
|Objective||Mathematics is of ever increasing importance to the Natural Sciences and Engineering. The key is the so-called mathematical modelling cycle, i.e. the translation of problems from outside of mathematics into mathematics, the study of the mathematical problems (often with the help of high level mathematical software packages) and the interpretation of the results in the original environment.|
The goal of Mathematics I and II is to provide the mathematical foundations relevant for this paradigm. Differential equations are by far the most important tool for modelling and are therefore a main focus of both of these courses.
|Content||1. Single-Variable Calculus:|
review of differentiation, linearisation, Taylor polynomials, maxima and minima, antiderivative, fundamental theorem of calculus, integration methods, improper integrals.
2. Linear Algebra and Complex Numbers:
systems of linear equations, Gauss-Jordan elimination, matrices, determinants, eigenvalues and eigenvectors, cartesian and polar forms for complex numbers, complex powers, complex roots, fundamental theorem of algebra.
3. Ordinary Differential Equations:
separable ordinary differential equations (ODEs), integration by substitution, 1st and 2nd order linear ODEs, homogeneous systems of linear ODEs with constant coefficients, introduction to 2-dimensional dynamical systems.
|Literature||- Thomas, G. B.: Thomas' Calculus, Part 1 (Pearson Addison-Wesley).|
- Bretscher, O.: Linear Algebra with Applications (Pearson Prentice Hall).
|Prerequisites / Notice||Prerequisites: familiarity with the basic notions from Calculus, in particular those of function and derivative.|
Mondays 12-14, Tuesdays 17-19, Wednesdays 17-19, in Room HG E 41.