Mathematical methods 2 for Computer Science engineers

Numerical methods in all parts where applicable.

• Complex numbers,
• Eigenvalues and eigenvectors.
• Power series, geometric series, Taylor series. series for exponential and trigonometric functions.
• Function of two and more variables, partial differentiation, extreme value problems.
• Logics: Statements, arguments, basic proof theory. Mathematical induction.
• Set theory and discrete functions.
• Number theory: Divisibility and congruence, RSA as application.
• Graph theory: Important types of graphs, graph isomorphy, trees. Graph theoretical algorithms, such as Prim's and Dijkstra's algorithms.
• Combinatorics: Counting results related to quantities, functions, and graphs.

Knowledge: The candidate should demonstrate knowledge of the following

• complex numbers.
• eigenvalues ​​and eigenvectors for square matrices.
• transformations in two dimensions.
• the concepts of series and convergence in general and geometric series in particular.
• Taylor series of important function types.
• functions of several variable and partial derivation.
• concepts in logic of statements.
• common forms of mathematical proofs, including mathematical induction.
• basic set theory.
• discrete functions.
• concepts and algorithms related to graphs, including trees and graphisomorphism.
• concepts, methods and results in number theory, modular calculus and cryptography.

Skills: The candidate can

• do calculations with complex numbers in normal and polar forms and can apply Euler's formula.
• apply characteristic polynomials in finding eigenvalues ​​and eigenvectors of a square matrix.
• use matrix multiplication to combine transformations in the plane
• apply eigenvalues ​​in practical applications
• use power series in approximations, knows Taylor's formula with residual term and can use the residual term to estimate the error in calculations
• derive partial derivatives and total differential and use them to linearize functions and to find stationary points of functions of two variables
• apply basic concepts, results and methods from the theory of statements and arguments, for example determine whether an argument is valid or invalid and determine whether two statements are equivalent
• construct simple mathematical proofs, including inductive proofs
• apply basic concepts and results related to set theory, discrete functions and can represent these in different ways
• apply basic concepts and results related to graphs, including paths in graphs and graphisomorphism
• apply algorithms to smaller examples,
• apply basic concepts and methods from number theory related to divisibility, including Euclid's method
• apply congruences calculations and perform RSA encryption and decryption

General competence: The candidate

• can use mathematics to model and solve theoretical and practical problems in situations relevant to their own field, in academic and professional contexts.
• The candidate can use computational and analytical tools to visualize and solve mathematical problems.

Lectures and exercises. Exercises will be based on assignments in a digital assessment system. Use of Python will also be included. Exercises and learning videos for self-study will be available as a supplement to the lectures.

Compulsory work: at least 75% of the exercises must be approved for admission to the exam. The number of obligatory assignments and weighting will be announced at the start of the semester.