Program Overview

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Program Overview

The University of Copenhagen offers a course titled "Geometry 1 (Geom1)" as part of its Bachelor's programs in Physics and Mathematics.

Course Description

The course covers various topics in geometry, including:

• Curves in space, cross product, dot product, curve length, and Frenet's 3-form.
• Parametrized surfaces: definition, examples (graphs, surfaces of revolution, level surfaces), tangent plane, parameter change, diffeomorphisms.
• Implicit function theorem and inverse function theorem.
• First fundamental form, inner geometry, local isometries.
• Integration on surfaces, including Stokes' theorem and Gauss' theorem.
• Curvature of surfaces, second fundamental form, principal curvatures, inner/Gaussian curvature, and outer/mean curvature.
• Theorema Egregium.
• Geodesic curves on curved surfaces.

Learning Objectives

Upon completion of the course, students will:

• Know that curves and surfaces in space can be treated mathematically, and that surprising results and deep insights can be obtained.
• Have expanded their knowledge of Euclidean spaces, particularly R^3.
• Have extended their knowledge of and familiarity with fundamental concepts from analysis and linear algebra, including implicitly defined functions.
• Be able to describe and calculate geometric quantities associated with parametrized curves in R^2 and R^3.
• Be able to describe and calculate geometric quantities associated with parametrized surfaces in R^3.
• Be able to describe and calculate geometric quantities associated with parametrized curves on parametrized surfaces in R^3.
• Be able to describe regular curves and regular surfaces in Euclidean spaces as level sets (i.e., solution sets to equations/systems of equations) or as graphs of (vector) functions (in several variables) or using parameterizations, and be able to rewrite (locally) between these three representation forms, and be able to use this (e.g., in the form of implicit differentiation) when calculating geometric quantities on curves and surfaces.
• Be able to prove statements about curves and surfaces in concrete examples.
• Have knowledge of the "inner geometry" of parametrized surfaces and be able to calculate associated quantities in special cases. Be able to master the proof of Theorema Egregium and its content.
• Be able to master the concept of local isometry between parametrized surfaces and geodesic curves on a parametrized surface.

Competences

The course provides a deeper understanding of fundamental mathematical objects such as functions, mappings, planes, spaces, distances, Euclidean spaces, and many advanced operations involving these, not least differentiation and linear algebra. Students learn that many previously learned definitions and constructions have geometric content, and conversely, that geometric objects such as surfaces in space can be described and analyzed mathematically. Finally, examples are seen where appropriate formulations lead to generalizations of concepts such as functions and differentiation, thereby achieving an even deeper understanding of the original structures and concepts.

Course Materials

Previous years have used the following material:

• Henrik Schlichtkrull: Curves and Surfaces. Notes from IMF, 2018.

Recommended Prerequisites

Linear Algebra (LinAlg) and Analysis 0 (An0).

Teaching Methods

The course consists of 5 hours of lectures and 4 hours of exercises per week for 7 weeks.

Workload

The estimated workload is:

• Category: Lectures, 35 hours
• Preparation (estimated), 139 hours
• Theoretical exercises, 28 hours
• Exam, 4 hours
• Total, 206 hours

Feedback

Feedback is provided in written and oral form, with ongoing feedback during the course.

Enrollment

Enrollment is through the self-service system on KUnet.

Examination

The examination is a written on-site exam with supervision, lasting 4 hours. The exam is graded on a 7-point scale, with external censorship. In case of a re-examination, the same format applies unless there are 10 or fewer enrolled, in which case the re-examination is a 30-minute oral exam with 30 minutes of preparation.

Course Information

• Language: Danish
• Course code: NMAA04013U
• Points: 7.5 ECTS
• Level: Bachelor
• Duration: 1 block
• Placement: Block 4
• Schedule group: B
• Course capacity: No limitation, unless enrolled during the late enrollment period (BA and KA) or as a merit or single-course student.

Study Board

The study board for Mathematics and Computer Science is responsible for the course.

Offering Institute

The Department of Mathematical Sciences offers the course.

Offering Faculty

The Faculty of Science offers the course.

Course Responsible

Niels Martin Møller is the course responsible.