MM512: Curves and Surfaces (5 ECTS)

STADS: 13001201

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. Fælles undervisning med MM842 Kurver og flader

Prerequisites:
The course cannot be chosen by students, who passed MM545.

Academic preconditions:
Students taking the course are expected to be familiar with: systems of linear equations, matrices, determinants,  vector spaces, scalar product and orthogonality, linear transformations,  eigenvectors and eigenvalues, polynomials, the concept of a function, real numbers, differentiation and integration of functions of one and several variables, vector calculus.

Course introduction
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), MM533 (Mathematical and Numerical Analysis)  and MM538 (Algebra and Linear Algebra). The course gives the prerequisites for more advanced courses in geometry.

The course is of high multidisciplinary value and gives an academic basis for a Bachelor Project in several core areas of Natural Sciences.

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. 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
2. apply these results to examples
3. formulate and present definitions, proofs and computations in a mathematically rigorous way

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

1. Curves and arc-length
2. Plane curves: signed curvature, the fundamental theorem, the isoperimetric inequality
3. Space curves: curvature and torsion, the fundamental theorem
4. 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.
5. 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. Pass/fail, internal marking by the teacher. (0 ECTS). (13001212).
2. Take-home exam at the end of the course. Grades according to the Danish 7-point scale, internal marking. (5 ECTS). (13001202).

The re-exam may differ from the ordinary exam.

Expected working hours
The teaching method is based on three phase model.
Intro phase: 28 hours
Skills training phase: 14 hours, hereof:
 - Tutorials: 14 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.