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
Email: dellamor@cp3.dias.sdu.dk

Timetable
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Comment:
Ubegrænset deltagerantal. Undervises fælles med MM507 Differentialligninger og MM512 Kurver og flader

Prerequisites:
None

Academic preconditions:
”Linear algebra” (MM505) (or ”Algebra and linear algebra” (MM538)) and ”Mathematical and numerical analysis” (MM533) should be known.

Course introduction
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.

Qualifications
Having completed the course successfully, the student can be expected to

• analyze and organize a domain or a situation with reference to mathematical modeling
• recognize problems which can be modeled by differential equations
• “read” the qualitative properties described by a model
• construct a mathematical model for a concrete problem
• find solutions analytical or numerical to ordinary differential equations by application of anappropriate tool
• to relate the solutions to the original problem
• work analytically with the following concepts, understand the geometric content of and connections between these ideas:
• arc-length, curvature and torsion for curves in the plane and three-dimensional space
• regularity of a parameterization
• principal, Gaussian and mean curvatures for a surface in three-dimensional space
• geodesic curves on parameterized surfaces in three-dimensions

Expected learning outcome
At the end of the course the student should be able 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
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

Meddeles ved kursets start

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:

1. Mandatory assignments. Evaluated by internal censorship on a pass/ fail basis (5 ECTS)
2. Oral examination. Evaluated by internal censorship by the danish 7 mark scale (5 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 Study phase: 76 hours

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.

Remarks
 In the first quarter taught together with the bachelor course MM534.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.