Course syllabus for Introductory mathematical analysis

Studies at the university begin with a two weeks non-compulsory introduction which is a repetition of the main features of high school mathematics. The introduction (about 30 hours) consists of: - information and diagnostic test - algebraic expressions - trigonometry - analytic geometry - functions. Literature: R. Pettersson: Förberedande kurs i matematik vid CTH

The course provides basic knowledge of mathematical analysis, which is necessary for most courses to follow at the Engineering Physics programme.

- understand the fundamental notions and definitions of mathematical analysis; - prove the most fundamental theorems within analysis of functions in one real variable; - use mathematical induction to prove identities and inequalities; - rewrite expressions which contain logarithms and the inverse of the trigonometric functions; - use a combination of standard limits in order to find other limits; - analyze functions in order to draw their graphs; - use standard methods to find the antiderivatives of some elementary functions; - use the main theorem of analysis to compute Riemann integrals; - apply Riemann integration on length, area and volume computation; - use comparison methods to determine convergence/divergence of improper integrals; - give proves of his/her own; - solve problems combining two or more of the above abilities.

Elementary set theory and logics. Proof by induction. Real numbers, absolute value, inequalities, Dedekind's intersection theorem for intervals, supremum / infimum. Functions, inverse functions. Exponential, power and logarithmic functions. Trigonometric functions and their inverse functions. Limits, continuity. Derivatives, derivation rules, differentials, the mean value theorem of differential calculus. Construction of curves, asymptotics, local extrema, maximal / minimal values, convex functions. Numerical solution of equations, iterative methods, method of Newton - Raphson (MATLAB, optional). Indefinite integrals, integration by parts, change of variables. The Riemann integral. Riemann sums and integration of continuous functions. The main theorem of integral calculus. The main theorem of analysis. The mean value theorem of integral calculus. Improper integrals. Numerical integration (MATLAB). Applications of integrals (volumes, length of curves etc).

Lectures, exercises, Matlab .

Litteratur: A. Persson, L.C. Böiers: Analys i en variabel (Studentlitteratur, Lund) Exercises: Analys i en variabel. Studentlitteratur.

Written paper which combines theory and problem solving.