MM512: Curves and Surfaces
Comment
Entry requirements
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 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
- 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
- Curves and arc-length
- Plane curves: signed curvature, the fundamental theorem, the isoperimetric inequality
- Space curves: curvature and torsion, the fundamental theorem
- 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.
- Geodesic curves and the equations describing them.
Literature
Examination regulations
Exam element a)
Timing
Tests
Mandatory assignments
EKA
Assessment
Grading
Identification
Language
Examination aids
ECTS value
Additional information
Exam element b)
Timing
Tests
Take-home exam at the end of the course
EKA
Assessment
Grading
Identification
Language
Examination aids
ECTS value
Additional information
Indicative number of lessons
Teaching Method
- preparation of exercises in study groups
- preparation of projects
- contributing to online learning activities related to the course