 COURSE SYLLABUS

Mathematical Analysis and Geometry 7.5 credits

Matematisk analys och geometri
Second cycle, M7028M
Version
Course syllabus valid: Autumn 2019 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.

Syllabus established
by Niklas Lehto 15 Feb 2019

Last revised
by Niklas Lehto 15 Feb 2019

 Education level Second cycle Grade scale G U 3 4 5 Subject Mathematics Subject group (SCB) Mathematics

Entry requirements

Some familiarity with basic concepts in mathematical analysis, such as induction, limits, derivatives, and linear algebra corresponding to the courses Calculus(M0047M) and Linear Algebra and Calculus (M0048M) or similar.

Selection

The selection is based on 20-285 credits

Course Aim
Mathematical analysis:
The course gives an introduction to the topic of analysis and real functions of one variable. Contrary to other analysis courses that focus on handling the tools, this course focuses on developing theorems that are used in analysis, starting from the real number axiom.

After reading the course, the student shall be able to
• formulate and understand mathematical proofs,
• explain used notions and their context
• give illustrative examples of the introduced notions and theorems
• apply the notions and theorems in problem solving
• discuss and critically review elementary preentation of the theory in corresponding courses in upper secondary school and universities,
• Apply mathematican notion and methods in the following topics:
• countable and uncountable sets,
• limints introduced via sequences,
• topology of real numbers for providing access to known properties of  functions (for example the mean value problem),
• differentiable functions.
• Show a ability to to integrate ideas from different topics,
• Do mathematical reasoning in a structured and logically consistent way.

Geometry:
After reading the course, the student shall be able to:
Use the ideas, symbols, representation forms, rules and algorithms in geometry that are covered by this course,  as well as:

• Applying geometric ideas and methods in the following topics:
• Identify properties of different geometric objects,
• To explain and use some definitions, postulates and use some definitions, postulates and theorems in Euclidean geometry,
• Construct and explain some geometric mappings,
• Perform simple computations in coordinate geometry,
• Explain some definitions and theorems in non-euclidean geometry,

Contents
Real numbers and functions, sequences and series of real numbers, functions and limits, continuous functions and differentiable functions.

Fundamental ideas and relations in geometric figures (congurence, similarities, the Pythagorean theorem, trigonometry in triangles). Postulates, definitions and theorems in Euclidean geometry. Geometric problem solving including proofs. Analytic and non-euclidian geometry.

Realization
The course is taught through lectures

Examination
A written exam is given at the end of the course. The grade obtained in this exam is the final grade of the course.

Transition terms
Cannot be included in the degree together with M7026M.

Examiner
Niklas Grip

Literature. Valid from Autumn 2019 Sp 1 (May change until 10 weeks before course start)
Steven G. Krantz, Real Analysis and Foundations, Chapman and Hall/CRC, 4 edition, December 2016.
Anders Tengstrand, Åtta kapitel om geometri, Studentlitteratur 2004.

Course offered by
Department of Engineering Sciences and Mathematics

Modules