The aim of this course is to provide the students with basic tools and competences regarding the analysis and applications of curves and surfaces in 3D. The main idea of the course is very well described by the following exerpt from the cover of the textbook: "Curves and surfaces are objects that everyone can see, and many questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques." One integral part of the course is to apply computer experiments with Maple in eachone of the three steps: To ask natural geometric questions, to formulate them in precise mathematical terms, and to answer them using techniques from calculus. The course also aims to give the students a firm background for further studies in the manifold engineering applications of differential geometric tools and concepts.

• Calculate the curvature, the torsion, and the Frenet-Serret basis for a given space curve.
• Apply the first and second fundamental form to analyze curves on surfaces in space.
• Recognize isometries and conformal maps between simple surfaces.
• Determine the principal curvatures and principal directions at every point of a given surface.
• Calculate the Gauss curvature and the mean curvature at every point of a given surface.
• Explain the invariant geometric significance of the Gauss curvature.
• Explain the connection between the second fundamental form, the Weingarten map and the principal curvatures and directions.
• Explain the connection between the total curvature, the normal curvature, and the geodesic curvature of a curve on a given surface.
• Recognize geodesic curves from data about the normal curvature and the total curvature of the curves.
• Apply the general surface theory to surfaces of revolution and to ruled surfaces.
• Apply the Gauss-Bonnet theorem to estimate the Euler characteristic of a given surface.
• Apply the general theory to a simple geometric problem and present the solution in the form of a report.

Curves and surfaces in 3D - with particular focus on metric and curvature properties. How to find the shortest path on a surface. How to bend a surface. How to calculate the number of holes of a compact surface. Individually chosen applications of differential geometry which span a diversity of possibilities including roler coaster constructions, geographic map projections, relativity (special or general), protein geometry, to mention but a few. The specific list of contents includes:
Curves with constant width, the Frenet-Serret 'apparatus' for curves in 3D, first and second fundamental forms for surfaces, Gaussian curvature and mean curvature, equiareal maps, isometries, surfaces of constant curvature, geodesics, fundamental results of Gauss, Codazzi-Mainardi, and Gauss-Bonnet.

Andrew Pressley: Elementary Differential Geometry, Springer, 2001.

Please contact the teacher for information on whether this course gives the student the opportunity to prepare a project that may participate in DTU´s Study Conference on sustainability, climate technology, and the environment (GRØN DYST). More information http://www.groendyst.dtu.dk/kursustilmelding.aspx