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Graduate courses

Departments' graduate courses for PhD-students.


Syllabus for

Academic year
The course has been discontinued 
TMA226 - Advanced calculus  
Matematisk fördjupning
Syllabus adopted 2019-02-19 by Head of Programme (or corresponding)
Owner: TKKEF
6,0 Credits
Grading: TH - Five, Four, Three, Fail
Education cycle: First-cycle
Major subject: Mathematics

Teaching language: Swedish
Application code: 54116
Open for exchange students: No
Only students with the course round in the programme plan

Module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0113 Examination 6,0c Grading: TH   6,0c   05 Jun 2020 pm J   12 Oct 2019 am SB_MU  


Inaktiv_Fredrik Inaktiv_Ohlsson


TMA225   Applied mathematics


In order to be eligible for a first cycle course the applicant needs to fulfil the general and specific entry requirements of the programme(s) that has the course included in the study programme.

Course specific prerequisites

  • Analysis and linear algebra:
  • Calculus (one and several variables)
  • Ordinary Differential Equations
  • Systems of linear equations
  • Matrix algebra and determinants
  • Linear spaces and eigenvalues
  • Least squares method
  • Modeling with Matlab


This course will provide deeper knowledge of mathematics, modeling and numerical calculations 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)

  • articulate and understand the meaning of definitions and theorems, as well as being able to prove some theorems.
  • perform simple mathematical reasoning and evidence on his/her own.
  • master important concepts in linear algebra and e.g. be able to regard a function as a vector in a vector space.
  • solve partial differential equations (PDEs) and certain integro-differential equations approximately (1-D finite element methods).
  • construct numerical algorithms and implement discrete solutions (in Matlab) and display the qualitative and quantitative benefits of the chosen approximation with both tables, charts and graphs.
  • determine convergence, absolute or conditional, of a series using appropriate convergence criteria.
  • determine if a function series is uniformly convergent.
  • determine the convergence zone of a power series.
  • determine the Fourier series of a periodic function.


General vector spaces (in particular over the complex numbers), the concepts of base and dimension (with examples from the solution space of a system of ordinary differential equations (ODEs)). Eigenvalues ​​and eigenvectors. Hermitian matrices and the spectral theorem. Inner product spaces. The Cauchy-Schwarz inequality. Projection in general vector spaces with applications to function spaces and Fourier series.

The course also deals with mathematical models in 1D processes where reaction and production as well as transport mechanisms, such as diffusion and convection, are included. Introduction to "Galerkin Finite Element Method" (FEM). Quadrature rules, interpolation and numerical differentiation. Convergence criteria for sequences, series, function sequences, function series (power series, Fourier series) are also included. Uniform convergence. Weierstrass M-test.


Lectures (4-6 hrs / week), exercises with "ticking" (2-4 hrs / week), laboratory work (0-2 hours / week) plus assignments related to it. Some material is not covered in the lectures, but is left for self-study. This material is, however, just as much part of the course. Tutorials, computer labs and assignments play an important role throughout the course and together results in an integration of the entire course content, from theory to practice.


The Course literature is specified at the course web page before the course starts.

Examination including compulsory elements

Written examination and assignments.

Published: Wed 26 Feb 2020.