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Graduate courses

Departments' graduate courses for PhD-students.


Course syllabus for

Academic year
MVE035 - Multivariable analysis
Course syllabus adopted 2022-02-02 by Head of Programme (or corresponding)
Owner: TKTFY
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, Engineering Physics

Teaching language: Swedish
Application code: 57135
Open for exchange students: No
Maximum participants: 130
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
0105 Examination 6,0 c Grading: TH   6,0 c    

In programs



Thomas Bäckdahl

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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

Linear algebra MVE670 and Real analysis TMA976 or equivalent courses.


The course will provide familiarity with the most basic theories in mathematical analysis in several variables and shed light on their applications in physics and technology.

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

The goal is to provide the students with the necessary mathematical tools in multivariable calculus and 3-dimensional vector analysis for subsequent courses in the Engineering Physics and Technical Mathematics programs. Among the most important learning outcomes are the following:

  • To understand the basic concepts of multivariable differential calculus, such as: partial derivative, differentiability, linearization, gradient, implicit and inverse function theorems
  • To be able to apply the chain rule to changes of variables in PDE
  • To be able to find and classify the stationary points of a multivariable function and apply this knowledge to the solution of optimization problems
  • To understand the definition of Riemann integral in arbitrary dimension
  • To be able to apply some basic techniques when computing multiple integrals, such as: inspection/symmetry, Fubini's theorem, change of variables, level surfaces
  • To be able to handle different parametrizations of curves and surfaces in 3-space, and understand the meaning of and be able to compute line and surface integrals
  • To understand Green's theorem in the plane, plus Gauss' and Stokes' theorems in 3-space and apply these to the computation of line and flux integrals
  • To acquire some basic knowledge of how the concepts of the course arise in physics, especially in mechanics and electromagnetism
  • To be able to differentiate under the integral sign


Functions of several variables. Partial derivatives, differentiability, the chain rule, directional derivative, gradient, level sets, tangent planes.
Taylor's formula for functions of several variables, characterization of stationary points.
Double integrals, iterated integration, change of variables, triple integrals, generalized integrals.
Space curves. Line integrals, Green's formula in the plane, potentials and exact differential forms.
Sufaces in R3, surface area, surface integrals, divergence and curl, Gauss' and Stokes' theorems.
Some physical problems leading to partial differential equations. Partial differential equations of the first order. Differentiating under the integral sign.
Functional determinants, inverse functions theorem, implicit functions. Extremal problems for functions of several variables, Lagrange's multiplier rule.


The teaching is organized into lectures and exercise sessions (the latter include demonstrations at the blackboard). There are voluntary items yielding bonus points:

  • Electronic tests.
There are also obligatory blackboard presentations. The students are divided into groups of 6. Each group presents a week of lecture material at the blackboard and writes a report.


A. Persson, L.-C. Böiers: Analys i flera variabler, Studentlitteratur, Lund.
Övningar till Analys i flera variabler, Institutionen för matematik, Lunds tekniska högskola.

L. Råde, B. Westergren: BETA - Mathematics Handbook, Studentlitteratur, Lund.

Examination including compulsory elements

A written examination.
Bonus point-yielding tests.
Obligatory blackboard presentations.

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.