HA080000 Analysis IV

Hiroshima University Syllabus

Japanese

Academic Year 2025Year School/Graduate School School of Science

Lecture Code HA080000 Subject Classification Specialized Education

Subject Name 解析学IV

Subject Name
（Katakana）カイセキガク４

Subject Name in
English Analysis IV

Instructor TAKIMOTO KAZUHIRO

Instructor
(Katakana) タキモト　カズヒロ

Campus Higashi-Hiroshima Semester/Term 2nd-Year,  Second Semester,  4Term

Days, Periods, and Classrooms (4T) Tues3-4,Fri5-6：SCI E210

Lesson Style Lecture Lesson Style
(More Details) Face-to-face, Online (on-demand)

Lectures on the blackboard

Credits 2.0 Class Hours/Week 4 Language of Instruction J : Japanese

Course Level 2 : Undergraduate Low-Intermediate

Course Area（Area） 25 : Science and Technology

Course Area（Discipline） 01 : Mathematics/Statistics

Eligible Students Sophomore (2nd year) students

Keywords Local maximum and local minumum of a multivarible function, Lagrange multiplier, Multiple integral, Iterated integral (Repeated integral), Change of variables, Improper integral.

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) Mathematics
（Knowledge and Understanding）
・Understanding classical basic theory which is a base of modern mathematics.  Being able to find and explain issues from specific events.
（Abilities and Skills）
・To acquire basic mathematical abilities (Ability to understand concepts, calculation ability, argumentation ability).

Class Objectives
/Class Outline This course, Analysis IV, which is the continuation of Analysis I, II and III, is provided for understanding the local maximum and local minimum of a multivariable function, the Riemann integral for multivariable function, and so on.

Class Schedule Lesson1 Local maximum and local minimum of multivariable function, I (Extremum point and Hessian matrix)
Lesson2 Local maximum and local minimum of multivariable function, II (Maximum value and minimum value)
Lesson3 Conditional extremum problem
Lesson4 Riemann double integrals on the rectangle, I (Definition of the definite integral)
Lesson5 Riemann double integrals on the rectangle, II (A continuous function must be integrable)
Lesson6 Riemann double integrals on the bounded set, I (Measure zero set in the sense of Jordan)
Lesson7 Riemann double integrals on the bounded set, II (Measurable set in the sense of Jordan)
Lesson8 Mid-term examination
Lesson9 Iterated integrals
Lesson10 Multiple integrals, area and volume
Lesson11 Change of varibles for multiple integrals, I (Meaning of Jacobian of diffeomorphism)
Lesson12 Change of varibles for multiple integrals, II (Proof of change of variables formula)
Lesson13 Improper multiple integrals, I (Definitions and properties)
Lesson14 Improper multiple integrals, II (Beta function and Gamma function)
Lesson15 Uniform convergence of the family of functions and differentiation under integral sign

Final lesson : Final examination

The proposed plan of the course may be changed in order for the students to deepen their understanding.

Text/Reference
Books,etc. Textbook:
[1] Takeru Suzuki, Yoshio Yamada, Yoshihiro Shibata and Kazunaga Tanaka, Rikokei-no-tameno Bibunsekibun I & II, Uchida Rokakuho, 2007.

Study-aid books:
[2] Ken-Ichi Shiraiwa, Kaisekigaku-nyumon, Gakujutsu Tosho Shuppan-sha, 1981.
[3] Nobuyuki Suita and Tsunehiko Shimbo, Rikokei-no-Bibunsekibun-gaku, Gakujutsu Tosho Shuppan-sha, 1987.
[4] Koji Kasahara, Bibunsekibun-gaku, Saiensu-sha, 1974.
[5] Kunihiko Kodaira, Kaiseki-nyumon I, Iwanami Shoten, 2003.
[6] Teiji Takagi, Kaiseki-gairon, 3rd Edition, Iwanami Shoten, 1983.

I strongly recommend that you have some books for exercises.

PC or AV used in
Class,etc. Handouts, Visual Materials

(More Details) I will hand out some documentations if necessary.

Learning techniques to be incorporated Post-class Report

Suggestions on
Preparation and
Review Lesson1--Lesson15 Review is necessary.

Requirements The course "Exercises in Analysis IV" is given for helping and deepening your understanding. It is strongly recommended that you take this course. Please attend "Exercises in Analysis IV" and solve many problems in order to understand calculus deeply.

Grading Method Reports (15 percents), Midterm examination and Final examination (85 percents).
If I assign some quizes, these scores are also considered.

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	2025Year	School/Graduate School	School of Science
Lecture Code	HA080000	Subject Classification	Specialized Education
Subject Name	解析学IV
Subject Name （Katakana）	カイセキガク４
Subject Name in English	Analysis IV
Instructor	TAKIMOTO KAZUHIRO
Instructor (Katakana)	タキモト　カズヒロ
Campus	Higashi-Hiroshima	Semester/Term	2nd-Year, Second Semester, 4Term
Days, Periods, and Classrooms	(4T) Tues3-4,Fri5-6：SCI E210
Lesson Style	Lecture	Lesson Style (More Details)	Face-to-face, Online (on-demand)
Lectures on the blackboard
Credits	2.0	Class Hours/Week	4	Language of Instruction	J : Japanese
Course Level	2 : Undergraduate Low-Intermediate
Course Area（Area）	25 : Science and Technology
Course Area（Discipline）	01 : Mathematics/Statistics
Eligible Students	Sophomore (2nd year) students
Keywords	Local maximum and local minumum of a multivarible function, Lagrange multiplier, Multiple integral, Iterated integral (Repeated integral), Change of variables, Improper integral.
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)	Mathematics （Knowledge and Understanding）・Understanding classical basic theory which is a base of modern mathematics. Being able to find and explain issues from specific events. （Abilities and Skills）・To acquire basic mathematical abilities (Ability to understand concepts, calculation ability, argumentation ability).
Class Objectives /Class Outline	This course, Analysis IV, which is the continuation of Analysis I, II and III, is provided for understanding the local maximum and local minimum of a multivariable function, the Riemann integral for multivariable function, and so on.
Class Schedule	Lesson1 Local maximum and local minimum of multivariable function, I (Extremum point and Hessian matrix) Lesson2 Local maximum and local minimum of multivariable function, II (Maximum value and minimum value) Lesson3 Conditional extremum problem Lesson4 Riemann double integrals on the rectangle, I (Definition of the definite integral) Lesson5 Riemann double integrals on the rectangle, II (A continuous function must be integrable) Lesson6 Riemann double integrals on the bounded set, I (Measure zero set in the sense of Jordan) Lesson7 Riemann double integrals on the bounded set, II (Measurable set in the sense of Jordan) Lesson8 Mid-term examination Lesson9 Iterated integrals Lesson10 Multiple integrals, area and volume Lesson11 Change of varibles for multiple integrals, I (Meaning of Jacobian of diffeomorphism) Lesson12 Change of varibles for multiple integrals, II (Proof of change of variables formula) Lesson13 Improper multiple integrals, I (Definitions and properties) Lesson14 Improper multiple integrals, II (Beta function and Gamma function) Lesson15 Uniform convergence of the family of functions and differentiation under integral sign Final lesson : Final examination The proposed plan of the course may be changed in order for the students to deepen their understanding.
Text/Reference Books,etc.	Textbook: [1] Takeru Suzuki, Yoshio Yamada, Yoshihiro Shibata and Kazunaga Tanaka, Rikokei-no-tameno Bibunsekibun I & II, Uchida Rokakuho, 2007. Study-aid books: [2] Ken-Ichi Shiraiwa, Kaisekigaku-nyumon, Gakujutsu Tosho Shuppan-sha, 1981. [3] Nobuyuki Suita and Tsunehiko Shimbo, Rikokei-no-Bibunsekibun-gaku, Gakujutsu Tosho Shuppan-sha, 1987. [4] Koji Kasahara, Bibunsekibun-gaku, Saiensu-sha, 1974. [5] Kunihiko Kodaira, Kaiseki-nyumon I, Iwanami Shoten, 2003. [6] Teiji Takagi, Kaiseki-gairon, 3rd Edition, Iwanami Shoten, 1983. I strongly recommend that you have some books for exercises.
PC or AV used in Class,etc.	Handouts, Visual Materials
(More Details)	I will hand out some documentations if necessary.
Learning techniques to be incorporated	Post-class Report
Suggestions on Preparation and Review	Lesson1--Lesson15 Review is necessary.
Requirements	The course "Exercises in Analysis IV" is given for helping and deepening your understanding. It is strongly recommended that you take this course. Please attend "Exercises in Analysis IV" and solve many problems in order to understand calculus deeply.
Grading Method	Reports (15 percents), Midterm examination and Final examination (85 percents). If I assign some quizes, these scores are also considered.
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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