2019 Geometry III

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Honda Nobuhiro 
Course component(s)
Lecture     
Day/Period(Room No.)
Tue3-4(H136)  Fri3-4(H136)  
Group
-
Course number
MTH.B331
Credits
2
Academic year
2019
Offered quarter
3Q
Syllabus updated
2019/3/18
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.
・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''.

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