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- Liberal arts and basic science courses
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Yamada Kotaro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Thr3-4(H112)
- Group
- -
- Course number
- MTH.B212
- Credits
- 1
- Academic year
- 2021
- Offered quarter
- 4Q
- Syllabus updated
- 2021/3/19
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
As a continuation of "Introduction to Geometry I" MTH.B211, the following items about surfaces in the Euclidean 3-space are introduced:
parametrized surface, the first fundamental form; the length, the angle, and the area, the second fundamental form, the principal curvatures, the Gaussian and mean curvatures, geodesics, the Gauss-Bonnet theorem, the fundamental theorem of surface theory.
The goal is an understanding fudamental materials of classical differential geometry of surfaces, and a preparation of modern differential geometry.
Student learning outcomes
The students will learn the basic matters of differential geometry of surfaces in the Euclidean 3-space. In particular
(1) To understand that the parametrization of surfaces and a notion of quantities which do not depend on parameters.
(2) To know the relationship between the shape of surfaces and curvatures.
(3) To know examples of global properties and local properties of surfaces.
(4) To confirm the theories by calculations on concrete examples.
Keywords
Differential Geometry, Surfaces, Gaussian cruvature, Mean curvature, the Gauss-Bonnet theorem.
Competencies that will be developed
| ✔ Specialist skills | ✔ Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Each lecture will be constructed under the hormworks assigned on the previous week, including problems related to the lecture and questions related to the material.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | The Gaussian curvature, the mean curvature (the first and second fundamental quantities) | Details will be provided during each class session. |
| Class 2 | Invariance under parameter changes (the first and second fundamental forms) | Details will be provided during each class session. |
| Class 3 | The Weingarten formula (the principal curvatures) | Details will be provided during each class session. |
| Class 4 | The Gauss formula (the Christoffel symbols) | Details will be provided during each class session. |
| Class 5 | The fundamental theorem for surface theory (Theorema Egregium) | Details will be provided during each class session. |
| Class 6 | Geodesics (the Gauss-Bonnet theorem) | Details will be provided during each class session. |
| Class 7 | Term Exam | 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)
Masaaki Umehara and Kotaro Yamada, DIfferential Geometry of curves and surfaces, World Scientific, 2017
Reference books, course materials, etc.
Sebastian Montiel y Antonio Ros, Curvas y superficie, Proyecto Sur, 1998.Details will be provided during each class session.
Manfredo P. do Carmo, Differenial Geoetry of Curves and Surfaces, Prentice-Hall Inc., 1976.
Assessment criteria and methods
Details will be explained in the course.
Related courses
- MTH.B211 ： Introduction to Geometry I
- LAS.M102 ： Linear Algebra I / Recitation
- LAS.M106 ： Linear Algebra II
- LAS.M101 ： Calculus I / Recitation
- LAS.M105 ： Calculus II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students is required to take the class MTH.B211 "Introduction to Geometry I", or to study the contents of the class.
Contact information (e-mail and phone) Notice : Please replace from "[at]" to "@"(half-width character).
kotaro[at]math.titech.ac.jp
Office hours
N/A.
Contact by E-mails, or by chats of the online course.
Other
Check the web page http://www.math.titech.ac.jp/~kotaro/class/2021/geom-2/index-jp.html and/or OCW, for details.
In addition to the subjects in "Related Courses“， the following cources are related to this subject:
Differential Equations I/II; Introduction to Topology I/II/III/IV; Geometry I/II/III
