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- Liberal arts and basic science courses
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Honda Nobuhiro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Tue3-4(H136) Fri3-4(H136)
- Group
- -
- Course number
- MTH.B331
- Credits
- 2
- Academic year
- 2018
- Offered quarter
- 3Q
- Syllabus updated
- 2018/3/20
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
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
Student learning outcomes
Students are expected to
・Understand the definition of differential forms.
・Be familiar with calculations of exterior differentiation.
・Underatand 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 that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | tensor product, exterior algebra | Details will be provided during each class session |
| Class 2 | differential forms on Euclidean spaces 1 | Details will be provided during each class session |
| Class 3 | differential forms on Euclidean spaces 2 | Details will be provided during each class session |
| Class 4 | vector analsyis on the 3-dimensional Euclidean space | Details will be provided during each class session |
| Class 5 | differential forms on manifolds | Details will be provided during each class session |
| Class 6 | exterior products of differential forms, tensor products of tensor fields | Details will be provided during each class session |
| Class 7 | tensor fields on manifolds, pull-back of differential forms by maps, the definition of exterior differentiation | Details will be provided during each class session |
| Class 8 | justification of the definition of exterior differentiation, representation of exterior differentiation in terms of vector fields | Details will be provided during each class session |
| Class 9 | de Rham cohomology | Details will be provided during each class session |
| Class 10 | orientaion on a manifold | Details will be provided during each class session |
| Class 11 | volume forms, criterion of orientability of manifolds, examples of non-orientable manifolds | Details will be provided during each class session |
| Class 12 | integration of differential forms with compact support | Details will be provided during each class session |
| Class 13 | (concrete) examples of integration of volume forms | Details will be provided during each class session |
| Class 14 | manifolds with boundary, and the orientation of its boundary | Details will be provided during each class session |
| Class 15 | Stokes' theorem, its applications and proof | Details will be provided during each class session |
Textbook(s)
None required
Reference books, course materials, etc.
None required
Assessment criteria and methods
Based on overall evaluation of the results for mid-term and final examinations. Details will be provided during class sessions.
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are expected to have passes ``Geometry I'' and ``Geometry II''.
