CC222705 Research Methods in Geometry

Hiroshima University Syllabus

Japanese

Academic Year 2024Year School/Graduate School School of Education

Lecture Code CC222705 Subject Classification Specialized Education

Subject Name 幾何学研究法I

Subject Name
（Katakana）キカガクケンキュウホウI

Subject Name in
English Research Methods in Geometry

Instructor TERAGAITO MASAKAZU

Instructor
(Katakana) テラガイト　マサカズ

Campus Higashi-Hiroshima Semester/Term 3rd-Year,  First Semester,  1Term

Days, Periods, and Classrooms (1T) Weds5-8：EDU L109

Lesson Style Lecture Lesson Style
(More Details) 　

We will have many discussion.

Credits 2.0 Class Hours/Week 　 Language of Instruction J : Japanese

Course Level 3 : Undergraduate High-Intermediate

Course Area（Area） 25 : Science and Technology

Course Area（Discipline） 01 : Mathematics/Statistics

Eligible Students Students of Faculty of Education , Especially those of the Program in Mathematics Education

Keywords Differential geometry, plane curves, space curves, surfaces in the 3-dim space

Special Subject for Teacher Education 　 Special Subject 　

Class Status
within Educational
Program
(Applicable only to targeted subjects for undergraduate students)
Criterion referenced
Evaluation
(Applicable only to targeted subjects for undergraduate students) Secondary School Mathematics Education
（Knowledge and Understanding）
・To understand basic knowledge of teaching contents of mathematic education.
（Abilities and Skills）
・To acquire and utilize the ability to think mathematically about teaching contents of mathematic education such as algebra, geometry, statistics and computer.

Class Objectives
/Class Outline Objectives and Theme
・Understand the contents of curves and surfaces that are taught in elementary and secondary mathematics education
・Understand major concepts and awareness of the issues in classical differential geometry.

Course Overview:
The main subject of this course is shifting from simple understanding to strict understanding in the classical differential geometry. This course will start with looking at an overview of the contents of curves and  surfaces that are covered in elementary and secondary education, and then deal with the theory of plane curves, theories of space curves, and theories of surfaces. The aim of this course is for students who wish to become a teacher to see essential ideas  behind mathematical strictness.

Class Schedule Lesson1: Plane curves and surfaces that are covered in high school mathematics

Lesson2: Plane curves and arc length parametrication

Lesson3: Curvature of plane curves

Lesson4: Space curves

Lesson5: Curvature and Torsion of Space Curves

Lesson6: Frenet-Serret Frames and Fundamental Theorem of Curves

Lesson7: Surface in the space, Tangent vectors and Normal vectors

Lesson8: First fundamental quantities of surface

Lesson9: Second fundamental quantities of surface
Lesson10: Normal curvature and Principal curvature

Lesson11：Gaussian curvature and Mean curvature

Lesson12: Gauss formula, Weingarten formula

Lesson13: Integrability conditions

Lesson14: Theorema egregium
Lesson15:  Riemannian geometry

There will be a final exam in Lesson16.

Text/Reference
Books,etc. There are a number of good books about differential geometry. You should borrow some from library and keep them at hand.

PC or AV used in
Class,etc. 　

(More Details)

Learning techniques to be incorporated 　

Suggestions on
Preparation and
Review Lesson1: The lesson will provide an overview of plane curves and surfaces that are covered in high school mathematics. Reading a high school textbook in advance will help your understand the lesson better.

Lesson2-3: The lessons will be about the theory of plane curve in classical differential geometry. There will be no difference from high school mathematics in that plane curve is represented by parameters.

Lesson4-6: Students will the theory of space curves following the theory of plane curves. The two theories have something in common such as using parametric representation. The theory of space curves requires a concept called torsion in addition to curvature.

Lesson7: This lesson will explain the accurate definition of surface in the space and introduce tangent vectors and normal vectors which serve as the basis for measuring a curve of a surface.

Leson8-11: The lessons will introduce the first and the second fundamental quantities.
The main goal is Gaussian Curvature which is a typical curvature.

Lesson12-13: The lessons will be about conditions to determine surface in the space, like a plane curve is essentially determined by curvature and a space curve is determined by curvature and torsion, lessons will explain conditions to determine a surface in the space.

Lesson14: We will study Theorema Egregium which is the most well known fact in the classical differential geometry．

Lesson 15: In the last lesson, we introduce Riemannian geometry by using the hyperbolic plane.

Requirements ・This course does not need students who choose to sit in the back row.
・Since many of the students wish to become a teacher, behavior that makes other students uncomfortable will not be tolerated.
・You should always think that how you can support learners, how you can make a lesson enriched, and how you can contribute to improvement of lessons."

Grading Method The determination of credits will be based on final exam 100%.

Practical Experience

Summary of Practical Experience and Class Contents based on it

Message

Other

Please fill in the class improvement questionnaire which is carried out on all classes.
Instructors will reflect on your feedback and utilize the information for improving their teaching.

Academic Year	2024Year	School/Graduate School	School of Education
Lecture Code	CC222705	Subject Classification	Specialized Education
Subject Name	幾何学研究法I
Subject Name （Katakana）	キカガクケンキュウホウI
Subject Name in English	Research Methods in Geometry
Instructor	TERAGAITO MASAKAZU
Instructor (Katakana)	テラガイト　マサカズ
Campus	Higashi-Hiroshima	Semester/Term	3rd-Year, First Semester, 1Term
Days, Periods, and Classrooms	(1T) Weds5-8：EDU L109
Lesson Style	Lecture	Lesson Style (More Details)
We will have many discussion.
Credits	2.0	Class Hours/Week		Language of Instruction	J : Japanese
Course Level	3 : Undergraduate High-Intermediate
Course Area（Area）	25 : Science and Technology
Course Area（Discipline）	01 : Mathematics/Statistics
Eligible Students	Students of Faculty of Education , Especially those of the Program in Mathematics Education
Keywords	Differential geometry, plane curves, space curves, surfaces in the 3-dim space
Special Subject for Teacher Education		Special Subject
Class Status within Educational Program (Applicable only to targeted subjects for undergraduate students)
Criterion referenced Evaluation (Applicable only to targeted subjects for undergraduate students)	Secondary School Mathematics Education （Knowledge and Understanding）・To understand basic knowledge of teaching contents of mathematic education. （Abilities and Skills）・To acquire and utilize the ability to think mathematically about teaching contents of mathematic education such as algebra, geometry, statistics and computer.
Class Objectives /Class Outline	Objectives and Theme ・Understand the contents of curves and surfaces that are taught in elementary and secondary mathematics education ・Understand major concepts and awareness of the issues in classical differential geometry. Course Overview: The main subject of this course is shifting from simple understanding to strict understanding in the classical differential geometry. This course will start with looking at an overview of the contents of curves and surfaces that are covered in elementary and secondary education, and then deal with the theory of plane curves, theories of space curves, and theories of surfaces. The aim of this course is for students who wish to become a teacher to see essential ideas behind mathematical strictness.
Class Schedule	Lesson1: Plane curves and surfaces that are covered in high school mathematics Lesson2: Plane curves and arc length parametrication Lesson3: Curvature of plane curves Lesson4: Space curves Lesson5: Curvature and Torsion of Space Curves Lesson6: Frenet-Serret Frames and Fundamental Theorem of Curves Lesson7: Surface in the space, Tangent vectors and Normal vectors Lesson8: First fundamental quantities of surface Lesson9: Second fundamental quantities of surface Lesson10: Normal curvature and Principal curvature Lesson11：Gaussian curvature and Mean curvature Lesson12: Gauss formula, Weingarten formula Lesson13: Integrability conditions Lesson14: Theorema egregium Lesson15: Riemannian geometry There will be a final exam in Lesson16.
Text/Reference Books,etc.	There are a number of good books about differential geometry. You should borrow some from library and keep them at hand.
PC or AV used in Class,etc.
(More Details)
Learning techniques to be incorporated
Suggestions on Preparation and Review	Lesson1: The lesson will provide an overview of plane curves and surfaces that are covered in high school mathematics. Reading a high school textbook in advance will help your understand the lesson better. Lesson2-3: The lessons will be about the theory of plane curve in classical differential geometry. There will be no difference from high school mathematics in that plane curve is represented by parameters. Lesson4-6: Students will the theory of space curves following the theory of plane curves. The two theories have something in common such as using parametric representation. The theory of space curves requires a concept called torsion in addition to curvature. Lesson7: This lesson will explain the accurate definition of surface in the space and introduce tangent vectors and normal vectors which serve as the basis for measuring a curve of a surface. Leson8-11: The lessons will introduce the first and the second fundamental quantities. The main goal is Gaussian Curvature which is a typical curvature. Lesson12-13: The lessons will be about conditions to determine surface in the space, like a plane curve is essentially determined by curvature and a space curve is determined by curvature and torsion, lessons will explain conditions to determine a surface in the space. Lesson14: We will study Theorema Egregium which is the most well known fact in the classical differential geometry． Lesson 15: In the last lesson, we introduce Riemannian geometry by using the hyperbolic plane.
Requirements	・This course does not need students who choose to sit in the back row. ・Since many of the students wish to become a teacher, behavior that makes other students uncomfortable will not be tolerated. ・You should always think that how you can support learners, how you can make a lesson enriched, and how you can contribute to improvement of lessons."
Grading Method	The determination of credits will be based on final exam 100%.
Practical Experience
Summary of Practical Experience and Class Contents based on it
Message
Other
Please fill in the class improvement questionnaire which is carried out on all classes. Instructors will reflect on your feedback and utilize the information for improving their teaching.

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