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Graduate courses

Departments' graduate courses for PhD-students.


Syllabus for

Academic year
MVE255 - Multivariable calculus and partial differential equations  
Flervariabelanalys och partiella differentialekvationer
Syllabus adopted 2020-02-10 by Head of Programme (or corresponding)
Owner: TKMAS
7,5 Credits
Grading: TH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Education cycle: First-cycle
Major subject: Mathematics

Teaching language: Swedish
Application code: 55136
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 7,5 c Grading: TH   7,5 c   31 May 2021 am J,  09 Oct 2020 am J,  27 Aug 2021 am J

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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

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 the 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.
  • know the basics of partial differential equations and boundary value problems
  • account for the basic ideas of the finite element method.
  • use the finite element method in MATLAB.
  • use MATLAB for solving problems.


The course is about derivatives and integrals in several variables and partial differential equations.  Equal emphasis is put on the three pillars: mathematical theory, analytic techniques, and numerical computation. 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. 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. Gauss divergence theorem.

Introduction to partial differential equations, derivation of the heat equation, boundary value problem, weak formulation, 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.


S. Larsson, A. Logg, A. Målqvist:  Analys och linjär algebra, del IV:  Flervariabelanalys och partiella differentialekvationer

Examination including compulsory elements

More detailed information of the examination will be given on the course web page before start of the course. 

Page manager Published: Thu 04 Feb 2021.