Program Overview

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Program Overview

The program in question is the Bachelor's program in Mathematics, specifically focusing on the course "Geometry 1 (Geom1)".

Course Description

The course "Geometry 1 (Geom1)" is part of the Bachelor's program in Mathematics and Physics. It covers various topics in geometry, including:

• Curves in space, cross product, dot product, curve length, and Frenet's 3-frame.
• 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 possibly 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, leading to surprising results and deep insights.
• Have expanded their knowledge of Euclidean spaces, especially R^3.
• Have extended their knowledge of and familiarity with fundamental concepts from analysis and linear algebra, including implicitly given functions.
• Be able to describe and calculate geometric quantities related to parametrized curves in R^2 and R^3.
• Be able to describe and calculate geometric quantities related to parametrized surfaces in R^3.
• Be able to describe and calculate geometric quantities related to parametrized curves on parametrized surfaces in R^3.
• 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.

Competencies

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.

Teaching Methods

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

Course Materials

Previous years have used the notes "Curves and Surfaces" by Henrik Schlichtkrull, IMF, 2018.

Prerequisites

Recommended academic prerequisites include Linear Algebra (LinAlg) and Analysis 0 (An0).

Assessment

The course assessment includes:

• A written exam with supervision, lasting 4 hours.
• Four mandatory assignments (written submissions), of which at least three must be passed to be eligible for the exam. There is an opportunity to resubmit the first three assignments once.

ECTS Credits

The course is worth 7.5 ECTS credits.

Level

The course is at the Bachelor's level.

Duration

The course lasts for one block.

Placement

The course is placed in Block 4.

Capacity

There is no limit to the number of students, unless you enroll during the late enrollment period (BA and KA) or as a merit or single-course student.

Responsible Institute

The course is offered by the Department of Mathematical Sciences.

Responsible Faculty

The course is part of the Faculty of Science.

Course Responsible

The course responsible is Niels Martin Møller.