MM512: Curves and Surfaces

Comment

This course is co-read with MM842.

Entry requirements

The course cannot be followed by students who have passed MM842.

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 :

handle complex and development-oriented situations in study and work contexts.

Give skills to:

apply the thinking and terminology from the subject's basic disciplines.
analyze and evaluate the theoretical and practical problems for the application of a suitable mathematical model.

Give knowledge and understanding of:

basic knowledge generation, theory and methods in mathematics.
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:

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

Content

The following main topics are contained in the course:

Curves in space: arc-length curvature and torsion, the fundamental theorem
Surfaces in space: 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, Guass’ Theorema Egregium.
Geodesics on surfaces and the equations describing them.

Literature

See itslearning for syllabus lists and additional literature references.

Examination regulations

Exam element a)

Timing

Autumn

Tests

Mandatory assignments

EKA

N300048112

Assessment

Second examiner: None

Grading

Pass/Fail

Identification

Full name and SDU username

Language

English

Examination aids

To be announced during the course

ECTS value

Exam element b)

Timing

January

Tests

Take-home exam at the end of the course

EKA

N300048102

Assessment

Second examiner: None

Grading

7-point grading scale

Identification

Full name and SDU username

Language

English

Examination aids

To be announced during the course

ECTS value

Additional information

The re-exam is changed to an oral exam if there are 9 or fewer students enrolled. The re-exam consists of several sets of questions that will be made available to the students in advance. Each set contains questions of different difficulty levels. At the exam one set will randomly be selected and the student must present the answers to the selected set of questions. The examiners can ask for details on the answers. This part takes roughly 20 minutes. At the end of the exam, the examiners can ask the student questions continuing in the same line or touching upon other topics discussed in the course. Total duration: 30 minutes.

Indicative number of lessons

42 hours per semester

Teaching Method

At the faculty of science, teaching is organized after the three-phase model ie. intro, training and study phase.
In order to enable students to achieve the learning objectives for the course, the teaching is organised in such a way that there are 58 lectures, class lessons, etc. on a semester.

These teaching activities are reflected in an estimated allocation of the workload of an average student as follows:

Intro phase (lectures, class lessons) - 28 hours
Training phase: 14 hours,

Activities during the study phase:

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

Teacher responsible

Name	E-mail	Department
Jørgen Ellegaard Andersen	jea@sdu.dk	Quantum Mathematics

Additional teachers

Name	E-mail	Department	City
William Elbæk Mistegård	wem@imada.sdu.dk	Quantum Mathematics

Timetable

Odense

Show full time table (start E23)
Show full time table (start F24)

Administrative Unit

Institut for Matematik og Datalogi (matematik)

Team at Educational Law & Registration

NAT

Offered in

Odense

Recommended course of study

Profile	Education	Semester	Offer period

Transition rules

Transitional arrangements describe how a course replaces another course when changes are made to the course of study.

If a transitional arrangement has been made for a course, it will be stated in the list.

See transitional arrangements for all courses at the Faculty of Science.