Foundations of Analysis, 7.5 credits

Aims and objectives

General aims for first cycle education

First-cycle courses and study programmes shall develop:

• the ability of students to make independent and critical assessments
• the ability of students to identify, formulate and solve problems autonomously, and
• the preparedness of students to deal with changes in working life.

In addition to knowledge and skills in their field of study, students shall develop the ability to:

• gather and interpret information at a scholarly level
• stay abreast of the development of knowledge, and
• communicate their knowledge to others, including those who lack specialist knowledge in the field.

(Higher Education Act, Chapter 1, Section 8)

Course objectives

Knowledge and comprehension
After having completed the course the student shall be able to

• discriminate between a mathematical proof and a general reasoning.

Proficiency and ability
After having completed the course the student shall be able to

• prove mathematical propositions within the scope of the present course using standard techniques and known theorems, and
• propose mathematical proofs with good structure and stringent notation.

Values and attitude
After having completed the course the student shall be able to

• give written and oral presentations of mathematical proofs which emphasize the crucial parts and are easy to follow.

Main content of the course

Real numbers. Number sequences and convergence, subsequences, Cauchy sequences and series. Topology and the Heine-Borel theorem. Continuous functions, compact functions and supremum. Uniform continuity. Differentiable functions, Riemann integrals, mean value theorems. Sequences of functions, absolute convergence and power series. Metric spaces.

Teaching methods

Teaching is done in the form of lectures.
If the course has only a few registered participants, the above-described teaching methods can be completely or partially replaced by supervision and self-study.

Students who have been admitted to and registered on a course have the right to receive tuition and/or supervision for the duration of the time period specified for the particular course to which they were accepted (see, the university's admission regulations (in Swedish)). After that, the right to receive tuition and/or supervision expires.

Examination methods

Examination, 7.5 credits (Code: A001)
Written and oral presentation of assignments.

For students with a documented disability, the university may approve applications for adapted or other forms of examinations.

For further information, see the university's local examination regulations (in Swedish).

Grades

According to the Higher Education Ordinance, Chapter 6, Section 18, a grade is to be awarded on the completion of a course, unless otherwise prescribed by the university. The university may prescribe which grading system shall apply. The grade is to be determined by a teacher specifically appointed by the university (an examiner).

In accordance with university regulations regarding grading systems for first and second-cycle courses (Vice-Chancellor’s decision ORU 2018/00929), one of the following grades shall be used: Fail (U), Pass (G) or Pass with Distinction (VG). For courses that are included in an international Master’s programme (60 or 120 credits) or offered to the university’s incoming exchange students, the grading scale of A-F shall be used. The vice-chancellor, or a person appointed by the vice-chancellor, may decide on exceptions from this provision for a specific course, if there are special grounds.

Grades used on course are Fail (F), Sufficient (E), Satisfactory (D), Good (C), Very Good (B) or Excellent (A).

Examination
Grades used are Fail (F), Sufficient (E), Satisfactory (D), Good (C), Very Good (B) or Excellent (A).

For further information, see the university's local examination regulations (in Swedish).

Specific entry requirements

Multivariable Calculus, 9 Credits.

For further information, see the university's admission regulations (in Swedish).

Transfer of credits for previous studies

Students who have previously completed higher education or other activities are, in accordance with the Higher Education Ordinance, entitled to have these credited towards the current programme, providing that the previous studies or activities meet certain criteria.

For further information, see the university's local credit transfer regulations (in Swedish).

Other provisions

All or part of the course may be given in English.

Reading list and other teaching materials