Course syllabus adopted 2026-02-20 by Head of Programme (or corresponding).
Overview
- Swedish nameFlervariabelanalys
- CodeMVE616
- Credits6 Credits
- OwnerTKAUT
- Education cycleFirst-cycle
- Main field of studyMathematics
- DepartmentMATHEMATICAL SCIENCES
- GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Course round 1
- Teaching language Swedish
- Application code 47130
- Open for exchange studentsNo
- Only students with the course round in the programme overview.
Credit distribution
Module | Sp1 | Sp2 | Sp3 | Sp4 | Summer | Not Sp | Examination dates |
|---|---|---|---|---|---|---|---|
| 0126 Examination 6 c Grading: TH | 6 c |
In programmes
Examiner
- Hossein Raufi
- Teaching Fellow, Algebra and Geometry, Mathematical Sciences
Eligibility
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
Specific entry requirements
The same as for the programme that owns the courseApplicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements
Course specific prerequisites
The student is expected to have prior knowledge corresponding to introductory courses in mathematical analysis (differential calculus and integral calculus), linear algebra, and introductory programming in Python.Aim
The aim of the course is to study the theory and applications of multivariable calculus, with a clear connection to analytical and numerical methods.The course develops the students ability to use both analytical and numerical methods, and to use programming in Python as a natural tool for modelling and computation. By combining theory, computation, and applications, the course strengthens both theoretical understanding and practical problem-solving skills, and provides a solid basis for further studies in mathematics and engineering.
Learning outcomes (after completion of the course the student should be able to)
Explain and apply basic concepts in the analysis of functions of several variables, such as sets, sequences, and convergence in 𝑅𝑛, limits and different types of continuity, and formulate and use the concepts of derivative, gradient, Jacobian and Hessian matrices, chain rule, and Taylors theorem to analyse and approximate such functions, for example by linearisation and numerical differentiation. Formulate problems in several variables as systems of equations and optimisation problems, use the gradient and directional derivatives to find stationary points, classify these using the Hessian matrix, and apply Newtons method
and basic methods for optimisation on compact sets and under constraints.
Formulate and compute double and triple integrals, use Fubinis theorem to separate integrals, perform change of variables with the Jacobian determinant (for example in polar, cylindrical, and spherical coordinates), and apply double and triple integrals to determine mean values, centres of mass, moments of inertia, and improper integrals.
Formulate and compute line and surface integrals of scalar and vector fields, use gradient, curl, and divergence, apply Greens theorem, Stokes theorem, and Gauss theorem (divergence theorem) in simple situations, and analyse
conservative and incompressible fields with interpretation in terms of work, circulation, and flux.
Formulate basic partial differential equations with associated boundary and
initial conditions.
Implement and use numerical algorithms in Python for computations within the scope of the course, and analyse, interpret, and communicate in writing modelling choices, computational results, and limitations in engineering applications.
initial conditions.
Implement and use numerical algorithms in Python for computations within the scope of the course, and analyse, interpret, and communicate in writing modelling choices, computational results, and limitations in engineering applications.
Content
Differential calculus in several variables
Sets, sequences, and convergence in 𝑅𝑛; limits and continuity (including uniform and Lipschitz continuity); differentiability, derivative, partial derivatives, gradient, Jacobian and Hessian matrices; chain rule; Mean Value Theorem and Taylors theorem; linearisation and numerical differentiation.
Equation solving and optimisation
Newtons method for systems; gradient and directional derivatives; stationary points and classification via the Hessian matrix; optimisation on compact sets and under constraints (Lagrange multiplier method); overview of the inverse and implicit function theorems.
Integral calculus in several variables
Double and triple integrals; Fubinis theorem; change of variables and the Jacobian determinant (polar, cylindrical, spherical coordinates); mean values, centres of mass, and moments of inertia; improper integrals.
Line and surface integrals
Line and surface integrals; work, circulation, flux; gradient, curl, and divergence; Greens theorem, Stokes theorem, and Gauss theorem; conservative and incompressible fields.
Partial differential equations
Basic concepts for PDEs.
Applications and programming
Python as a computational tool; implementation, testing, and verification; for example training neural networks as an application of multivariable calculus and the gradient method.
Organisation
Teaching consists of lectures and exercise classes in smaller groups. More detailed information is provided on the course web page before the start of the course.Literature
S. Larsson, A. Logg, A. Målqvist, MATEMATISK ANALYS & LINJÄR ALGEBRA (IV): Flervariabelanalys och partiella differentialekvationerExamination including compulsory elements
Assessment is by a written examination. Assignments or short quizzes that may give bonus points towards the examination may be included.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 about disability study support.