COURSE SYLLABUS

Principles of Mathematical Analysis 7.5 credits

Analysens grunder
Second cycle, M7026M
Version
Course syllabus valid: Autumn 2021 Sp 1 - Present
The version indicates the term and period for which this course syllabus is valid. The most recent version of the course syllabus is shown first.

 Education level Second cycle Grade scale U G VG * Subject Mathematics Subject group (SCB) Mathematics

Entry requirements

U0021P - Mathematics and Learning for Teachers part 2 or equivalent.

Selection

The selection is based on 20-285 credits

Course Aim

After finishing the course, the student shall be able to:

Knowledge and understanding

• Explain the central concepts, definitions and theorems in the Course contents and how they are related.
• Give illustrating examples for the introduced concepts and theorems.

Skills and abilities

• Prove central theorems introduced in the course.
(Except for some particularly complicated proofs, where it is enough to be able to read the proofs and explain how they combine different mathematical concepts and results.)
• Explain a mathematical line of argument in a structured and logically coherent way.
• Apply introduced concepts and theorems to problem solving. Some typical examples:
- Decide if a given set is countable.
- Confirm the limit of a function or sequence using the definition of limit.
- Compute a given limit of a function using l’Hospital’s rule.
- Decide if a given function is Riemann integrable and explain why.
- Decide if a given series converges and explain why.
- Compute the limit of certain series.
- Decide if a given sequence of functions converges pointwise and/or uniformly.

Assessment and attitude

• Future math teachers finishing the course shall be able to discuss and critically examine elementary descriptions of the theory in corresponding courses at upper-secondary school and university level.
• Future math teachers finishing the course shall be able to give useful supervision for unusually talented pupils who need guidance and help outside the  ordinary curriculum. (This also requires pedagogic skills that are not examined in this course, but in other courses and in ”pedagogic strips” for the Master programme in Secondary Education.)

Contents
The course covers the following topics:
• Number systems.
• Harmonisc, geometric and arithmetiv means.
• Countable and uncountable sets.
• Sequences and limits of sequences and functions.
• Topology of the real numbers for  av reella tal for showing known properties of  functions, such as mean value theorems.
• Uniform continuity.
• Lipschitz continuity.
• Differentiable functions, mean value theorems and l’Hospital’s rules.
• Riemann integrals, definition and properties.
• Discontinuous functions that are  integrerbble under special conditions (for example functions with jump discontinuities).
• Lebesgue measure 0 and the Lebesgue theorem about the set of Riemann integrable functions.
• Infite sums (series) of real numbers and functions. Convergence criteria and construction of contionuous functions that are nowhere differentiable.
• Uniform convergence.

Realization
Each course occasion´s language and form is stated and appear on the course page on Luleå University of Technology's website.
The teaching currently consists of lectures and homework problems for the students. In the future, this can be extended with for example problem solving classes or seminars where the students are more active, but that would first require making recorded lectures for  setting free time for such activities.

Examination
If there is a decision on special educational support, in accordance with the Guideline Student's rights and obligations at Luleå University of Technology, an adapted or alternative form of examination can be provided.
Written exam with grading scale U, G, VG.

Remarks
The course is compulsory for the Master programme in Secondary Education (for future math teachers in the upper-secondary school), but also chosen by interested students from other programmes (especially engineering programmes).

Important parts of the underlying  theory is left out in the courses mentioned under ”Entry requirements” above, and the chosen courses alsoomit important topics like series och l’Hospitals rule.

This course fills those gaps, which is both an important foundation for continued math studies, as well as for the goals under  ”Assessment and attitude”.

Examiner
Niklas Grip

Literature. Valid from Autumn 2021 Sp 1 (May change until 10 weeks before course start)
Parzynski, W.R. and Zipse, P.W. (1987). Introduction to Mathematical Analysis (first seven chapters), McGraw-Hill.
or
Steven G. Krantz, Real Analysis and Foundations, Chapman and Hall/CRC, 4 edition, December 2016.
Lecture material (about 300 pages) in Canvas that closely follows the Parzynskki and Zipse book, but also gives references to the corresponding sections in Krantz’ book.
Homework problems and suggested solutions in Canvas.

Course offered by
Department of Engineering Sciences and Mathematics

Modules