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Graduate courses

Departments' graduate courses for PhD-students.


Course syllabus for

Academic year
TMA227 - Advanced calculus  
Matematisk fördjupning
Course syllabus adopted 2021-02-26 by Head of Programme (or corresponding)
Owner: TKKEF
6,0 Credits
Grading: TH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Education cycle: First-cycle
Main field of study: Mathematics

Teaching language: Swedish
Application code: 54116
Open for exchange students: No
Maximum participants: 50
Only students with the course round in the programme overview
Status, available places (updated regularly): Yes

Module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0119 Examination 6,0 c Grading: TH   6,0 c   07 Oct 2022 pm J

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Tony Johansson

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General entry requirements for bachelor's level (first cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

The same as for the programme that owns the course.
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites

  • Analysis and linear algebra:
  • Calculus (one and several variables)
  • Ordinary differential equations
  • Systems of linear equations
  • Matrix algebra and determinants
  • Linear Euclidean spaces and eigenvalues
  • The least squares method


This course will provide a deeper knowledge of mathematics in such a way that further studies within the Kf program is facilitated. Special care is taken to provide the knowledge needed for more advanced courses in mathematics and physics.

Learning outcomes (after completion of the course the student should be able to)

  • formulate, and explain the meaning of, relevant concepts, definitions and theorems
  • prove some relevant fundamental theorems
  • independently construct simple mathematical arguments and proofs
  • master important concepts in linear algebra and for example be able to treat a function as a vector in a vector space
  • analyze the convergence of sequences and solve linear difference equations
  • determine convergence, absolute or conditional, of a series using appropriate convergence criteria
  • determine if a function series is uniformly convergent
  • determine the convergence region of a power series
  • apply results concerning the interchanging of limits, and term-wise integration and differentiation
  • determine the Fourier series of a periodic function


General vector spaces and subspaces, the concepts linearly independent vectors, basis and dimension. Linear transformations. The dimension theorem. Orthogonality and inner product spaces. The Cauchy-Schwarz inequality. Orthogonal projection in general vector spaces with applications to function spaces and Fourier series.
Sequences and difference equations, series, sequences and series of functions. Convergence criteria. Uniform convergence of sequences and series. Interchange of limit procedures. Weierstrass majorant test. Applications to power series and Fourier series.


The teaching is organized in the form of lectures and tutorials. Bonus points may be employed. Some material is not covered in the lectures, but left for self-study. This material is, however, just as much part of the course. The tutorials play an important role throughout the course in the integration of the entire course content, from theory to practice.


The course literature is specified on the course web page before the course starts.

Examination including compulsory elements

Written exam at the end of the course.
During the course, there may be assignments for bonus points to the written exam, e.g. in the form of quizzes or written and/or oral presentation of problem-solving assignments. More detailed information about the examination and information regarding any bonus assignments for each course instance is provided on the course web page before the start of the course.

The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers on educational support due to disability.

Page manager Published: Thu 04 Feb 2021.