2021 Faculty Courses School of Science Undergraduate major in Mathematics
Geometry III
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Toshiaki Hattori
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue (S323) / 3-4 Fri (S323)
- Class
- -
- Course Code
- MTH.B331
- Number of credits
- 200
- Course offered
- 2021
- Offered quarter
- 3Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The aim of this course is to familiarize the students with basic notions and properties on differential forms on differentiable manifolds. The contents of this course is as follows: tensor algebras and exterior algebras, the definition of differential forms, exterior differentiation, de Rham cohomology, orientations of manifolds, integration of differential forms, Stokes' theorem.
Course description and aims
Students are expected to:
- Understand the definition of differential forms
- Be familiar with calculations of exterior differentiation
- Understand the definition of de Rham cohomology
- Be able to use Stokes' theorem
Keywords
tensor algebras, exterior algebras, differential forms, exterior differentiation, de Rham cohomology, orientation, volume forms, integration of differential forms, manifolds with boundary, Stokes' theorem
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | tensor algebras |
Details will be provided during each class session |
| Class 2 | alternating forms |
Details will be provided during each class session |
| Class 3 | exterior algebras |
Details will be provided during each class session |
| Class 4 | tensor fields and differential forms on manifolds |
Details will be provided during each class session |
| Class 5 | pull-back of differential forms by maps |
Details will be provided during each class session |
| Class 6 | the definition of exterior differentiation, examples |
Details will be provided during each class session |
| Class 7 | justification of the definition of exterior differentiation |
Details will be provided during each class session |
| Class 8 | de Rham cohomology |
Details will be provided during each class session |
| Class 9 | orientaion on a manifold |
Details will be provided during each class session |
| Class 10 | volume forms, criterion of orientability of manifolds, examples of non-orientable manifolds |
Details will be provided during each class session |
| Class 11 | integration of differential forms |
Details will be provided during each class session |
| Class 12 | (concrete) examples of integration of differential forms |
Details will be provided during each class session |
| Class 13 | manifolds with boundary, and the orientation of its boundary |
Details will be provided during each class session |
| Class 14 | Stokes' theorem, its applications and proof |
Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required
Reference books, course materials, etc.
None required
Evaluation methods and criteria
Examination and assignment. Details will be provided during class sessions.
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
Prerequisites
Students are expected to have passes ``Geometry I'' and ``Geometry II''.
Other
We use OCW-i.