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Graduate courses

Departments' graduate courses for PhD-students.


Syllabus for

Academic year
MVE255 - Calculus in several variables  
Matematisk analys i flera variabler
Syllabus adopted 2019-02-21 by Head of Programme (or corresponding)
Owner: TKMAS
7,5 Credits
Grading: TH - Five, Four, Three, Fail
Education cycle: First-cycle
Major subject: Mathematics

Teaching language: Swedish
Application code: 55155
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
0108 Examination 7,5c Grading: TH   7,5c   02 Jun 2020 pm J,  11 Oct 2019 am SB_MU   28 Aug 2020 am J

In programs



Axel Flinth

  Go to Course Homepage


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

Calculus in one variable, linear algebra and programming in Matlab.


The purpose of the course is to, together with the other math courses in the program, provide a general knowledge in the mathematics required in further studies as well as in the future professional career.

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

  • account for meaning of the concepts of calculus in several variables.
  • account for the relations between these concepts and to use them in problem solving.
  • implement the Newton method and the method of gradients in MATLAB functions. 
  • solve optimization problems with constraints.
  • account for the basic ideas of the finite element method.
  • use the finite element method in MATLAB.
  • use MATLAB for solving problems.


The space Rn, open, closed, compact sets.
Functions from Rn to Rm, curves and surfaces.
Limits, continuity, differentiability, the chain rule.
Partial derivative, linearization, Jacobi matrix, gradient, tangent plane, directional derivative, differentials.
Functional matrix, functional determinant.
Numerical solution of systems of nonlinear equations.

Extreme values, optimization in compact sets, optimization with constraints. Numerical optimization: Newton's method and the method of gradients.

Double and triple integrals. Polar and spherical coordinates, substitution of variables. Computation of volume, center of mass, area of a curved surface.

Line integral. Green's formula.

Introduction to partial differential equations: Laplace and Poisson equations, finite element method.

MATLAB applications from mechanics.


Instruction is given in lectures and classes. More detailed information will be given on the course web page before the start of the course.


T. Kolsrud, T. Lindström och K. Hveberg, Flervariabelanalys med linjär algebra, Prentice Hall 2012. 
S. Larsson, Randvärdesproblem och finita elementmetoden, kompendium, 2014.

Examination including compulsory elements

More detailed information of the examination will be given on the course web page before start of the course.
Examples of assessments are:
-selected exercises are to be presented to the teacher orally or in writing during the course,
-other documentation of how the student's knowledge develops,
-project work, individually or in group,
-written or oral exam during and/or at the end of the course.
-problems/exercises are to be solved with a computer and presented in writing and/or at the computer.

Published: Wed 26 Feb 2020.