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2024　Geometry III

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Hattori Toshiaki 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
-
Group
-
Course number
MTH.B331
Credits
2
Academic year
2024
Offered quarter
3Q
Syllabus updated
2024/3/14
Lecture notes updated
-
Language used
Japanese
Access Index

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
- 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 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 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.

Out-of-Class Study Time (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　　　　　　

Assessment criteria and methods

Examination and assignment. 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''.

Other

None in particular.

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