MM545: Ordinary differential equations and geometry (10 ECTS)

STADS: 13012801

Level
Bachelor course

Teaching period
The course is offered in the autumn semester.

Teacher responsible
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Timetable
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Comment:
Samlæses i ugerne 36-44 med MM547

Prerequisites:
The course cannot be chosen by students, who followed MM547, MM507, MM512, or MM534.

Academic preconditions:
Students taking the course are expected to:

• Have knowledge of how to implement algorithms as computer programs and compute numerical approximations to mathematical problems that don't allow a closed form solution.
• Be familiar with: systems of linear equations, matrices, determinants,  vector spaces, scalar product and orthogonality, linear transformations,  eigenvectors and eigenvalues, diagonalisation, polynomials, the concept of a function, real and complex numbers, differentiation and integration of functions of one and several variables, vector calculus.

Course introduction
The purpose of the course is to introduce modelling of problems from science and engineering by ordinary differential equations and to analyse and solve these equations. The course will introduce analytic techniques to deal with parameterized curves and surfaces in three dimensions and give the students methods to visualize the geometric results obtained.

The course builds on the knowledge acquired in the courses MM536 (Calculus for Mathematics) and one of MM505 (Linear Algebra) or MM538 (Algebra and Linear Algebra). The gives an academic basis for a Bachelor Project in several core areas of Natural Sciences, that requires mathematical modelling as well as more advanced courses in differential equations.

In relation to the competence profile of the degree it is the explicit focus of the course to:

• Give the competence to :
1. handle complex and development-oriented situations in study and work contexts.
• Give skills to:
1. apply the thinking and terminology from the subject's basic disciplines.
2. analyze and evaluate the theoretical and practical problems for the application of a suitable mathematical model.
• Give knowledge and understanding of:
1. basic knowledge generation, theory and methods in mathematics.
2. how to conduct analyses using mathematical methods and critically evaluate scientific theories and models.

Expected learning outcome
The learning objectives of the course are that the student demonstrates the ability to:

1. formulate a differential equation as a model for a simple problem
2. solve differential equations by methods taught in the course
3. find steady states and analyse the asymptotic behaviour of simple systems of differential equations
4. reproduce definitions and results, together with their proofs, in the geometry of plane- and space-curves and of surfaces in space, within the scope of the course's syllabus
5. apply these results to examples
6. formulate and present definitions, proofs and computations in a mathematically rigorous way

Subject overview
The following main topics are contained in the course:

1.1. First order differential equations and mathematical models.
1.2. Slope fields and initial value problems.
1.3. Euler's approximation.
1.4. Existence and uniqueness, Picard-Lindelöf theorem (as application of fixed point theorem).
1.5. Gronwall's Lemma and the convergence of Euler's method.
1.6. Analytic tools: integrating factors, separation of variables, and exact equations.
2.1. Systems of first order linear differential equations, and linear higher order differential equations: fundamental solutions, the solution space.
2.2. The Wronskian, Abel's theorem.
2.3. Analytic tools: undetermined coefficients and the variation of parameters.
3. Curves and arc-length
4. Plane curves: signed curvature, the fundamental theorem, the isoperimetric inequality
5. Space curves: curvature and torsion, the fundamental theorem
6. Parameterized surfaces: regular patches, the tangent space, graphs, surfaces of revolution, normal curvature, geodesic curvature, the first and second fundamental forms, principal curvatures, Gaussian curvature, mean curvature.
7. Geodesic curves and the equations describing them.

Literature
There isn't any litterature for the course at the moment.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:

1. Mandatory assignments. Internal marking by teacher on a pass/fail basis (5 ECTS).
2. Oral examination. Evaluated by internal marking by the Danish 7-mark scale (5 ECTS).
3. Mandatory assignments. Internal marking by teacher on a pass/fail basis (0 ECTS).

Reexam in the same exam period or immediately thereafter. The mode of a reexamination may differ from the mode of the ordinary examination.

Expected working hours
The teaching method is based on three phase model.
Intro phase: 56 hours
Skills training phase: 28 hours, hereof:
 - Tutorials: 28 hours

Educational activities

Educational form
Activities during the study phase:

• preparation of exercises in study groups
• preparation of projects
• contributing to online learning activities related to the course

Language
This course is taught in Danish or English, depending on the lecturer. However, if international students participate, the teaching language will always be English.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.