Established by: Faculty Board of Science and Technology, 2022-03-14
Mathematical models are essential tools in modern science. In this course, we use tools from analysis, linear algebra, and topology to introduce a model that has not only been used in mathematics but also classically is used in physics such as e.g. in Hamiltonian mechanics and Einstein's theory of relativity. Other applications are found in chemistry, economics, engineering, computer graphics and computer vision, as well as recently in statistical learning and machine learning. Some of the key concepts covered in the course are manifolds, tangent spaces, vector fields, tensors, and differential forms. The course concludes with Stokes' theorem on manifolds.
Expected learning outcomes
For a passing grade, the student must be able to
Knowledge and understanding
account for the theory of differential manifolds
account for tangent spaces, vector fields, vector bundles, and their dual counterparts
account for tensors and differential forms
formulate and prove central theorems
Skills and abilities
apply tensors for metrics
integrate over differentiable manifolds
apply theorems for problem solving
The course requires 90 ETCS including a course in Real Analysis and a course in Topology or equivalent. Proficiency in English and Swedish equivalent to the level required for basic eligibility for higher studies
Form of instruction
The teaching on this course consists of lectures and exercise classes.
The course is examined through written exams. For the whole course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG). In order to pass the course, a student must have passed all parts of the assessment.
Deviations from the syllabus examination form can be made for a student who has a decision on pedagogical support due to disability. Individual adaptation of the examination form shall be considered based on the student's needs. The examination form is adapted within the framework of the expected learning outcomes of the course syllabus. At the request of the student, the course coordinator, in consultation with the examiner, must promptly decide on the adapted examination form. The decision shall then be communicated to the student.
A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.
Credit transfer All students have the right to have their previous education or equivalent, and their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications.
In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics.
In the event that the syllabus ceases to apply or undergoes major changes, students are guaranteed at least three examinations (including the regular examination opportunity) according to the regulations in the syllabus that the student was originally registered on for a period of a maximum of two years from the time that the previous syllabus ceased to apply or that the course ended.