EPN-V2

M1GMT1200 Matematikk 1, emne 2 Emneplan

Engelsk emnenavn
Mathematics 1, Subject 2
Studieprogram
Grunnskolelærerutdanning for trinn 1-7
Omfang
10.0 stp.
Studieår
2025/2026
Programplan
Emnehistorikk

Innledning

Fagplanen tilhørende dette emnet er lagt på emne M1GMT1100 Matematikk 1, emne 1.

Forkunnskapskrav

None.

Læringsutbytte

Students who complete the course are expected to have the following learning outcomes, defined in terms of knowledge, skills and general competence:

Knowledge

On successful completion of the course, the student:

  • knows how mathematical models can be derived from facts and first principles.
  • has a repertoire of methods to solve and/or analyse both ordinary differential equation (ODE) systems and certain partial differential equations (PDEs).
  • is able to apply analytical and/or numerical solution methods for PDEs to models of heat transfer, wave propagation and diffusion-convection and discuss the relevance of these models to real-world phenomena.
  • is able to construct and develop relevant models and discuss the validity of the models.

Skills

On successful completion of the course, the student can:

  • can determine steady states of ODE systems and use linear approximation to elucidate the stability properties of these states.
  • can solve and/or analyse selected PDE models.
  • is able to implement and use some numerical methods for solving relevant PDEs.
  • can devise the solution of certain composite quantitative problems.
  • can disseminate results and findings in an accessible manner – both orally and in writing.

General competence

  • is aware of the usefulness and limitations of mathematical modelling as well as of pitfalls frequently encountered in modelling and simulation.
  • is able to discuss properties of a system using the equations of the mathematical model that describes the system.
  • can explain and use numerical methods, know their strengths and weaknesses and interpret results of numerical simulations.

Innhold

Introductory module:

  • Principles of modelling and derivation of mathematical models
  • Analysis of ordinary differential equations (ODEs)
  • Linear partial differential equations (PDEs)
  • Prominent results from functional analysis and their application to ODEs and PDEs
  • Numerical methods for computing of solutions of PDEs

Functional analysis:

  • Completeness for normed spaces
  • Hilbert spaces, compact and diagonalisable operators
  • Theory of topological vector spaces
  • Test functions, distributions and the Fourier transform
  • Sobolev spaces and fundamental solutions of partial differential equations

Biosystems:

  • Mathematical models for biological systems
  • Analytical and numerical methods for simulation of system response
  • Actuators and sensors for stimulation and measurements of biological systems
  • Interaction of biological and measurement system

Modern physics:

  • Monte Carlo techniques
  • Splines and other expansion techniques
  • Applications of expansions in spherical harmonics
  • Numerical problems in general relativity and quantum physics
  • Manifolds with geometric structures central to physics and engineering.

Within all specializations, the content may be adjusted to accommodate for the research area of each PhD candidate.

Arbeids- og undervisningsformer

Se fagplanen.

Arbeidskrav og obligatoriske aktiviteter

The teaching is organised as sessions where the subject material is presented, and as sessions where the students solve problems using analytical and/or numerical methods. Between these sessions, the students should work individually with literature studies and problem solving.

In the last, specialised part, the students are required to complete and present a rather extensive individual project involving theoretical and practical/implementational aspects.

Vurdering og eksamen

The following required coursework must be approved before the student can take the exam:

  • Completion of an extensive individual project in the specialised module.

Hjelpemidler ved eksamen

The student's own project.

Vurderingsuttrykk

Pass or fail.

Sensorordning

Two examiners. External examiner is used periodically.

Opptakskrav

A thorough knowledge of advanced calculus, including ordinary and partial differential equations. It is a great advantage if students are familiar with linear algebra and Fourier and Laplace transform theory. In terms of programming, some experience in implementing various numerical methods, including schemes for solving partial differential equations is recommended. Some knowledge of mathematical analysis, modern physics or physiology is recommended, depending on their specialisation.