MM512: Curves and Surfaces

The course builds on the knowledge acquired in the courses MM536 (Calculus for Mathematics), or equivalent.

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.

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 gives the prerequisites for more advanced courses in geometry.

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

The following main topics are contained in the course:

1. Curves in space: arc-length curvature and torsion, the fundamental theorem
2. 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.
3. Geodesics on surfaces and the equations describing them.

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