HA095000 Exercises in Analysis IV

Hiroshima University Syllabus

Japanese

Academic Year 2025Year School/Graduate School School of Science

Lecture Code HA095000 Subject Classification Specialized Education

Subject Name 解析学IV演習

Subject Name
（Katakana）カイセキガク４エンシユウ

Subject Name in
English Exercises in Analysis IV

Instructor KAWASHITA MISHIO

Instructor
(Katakana) カワシタ　ミシオ

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

Days, Periods, and Classrooms (4T) Mon1-2,Thur3-4：SCI E209

Lesson Style Seminar Lesson Style
(More Details) Face-to-face

Blackboards and projectors are used. Sometime, prints are given.

Credits 1.0 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 students of department of mathematics in faculty of science

Keywords multiple integral, iterated integral (repeated integral), change of order of integration, aria of sets and measure zero sets in the sense of Jordan, change of variables, improper integral, volume, surface aria

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 is to help you understand the contents of "Analysis IV." The course is given by exercise style.

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

It is possible to change the order of the contents or the contents themselves. Perhaps, some of the quizzes in the class or the reports are given.

The proposed plan of the course may change when it is better for participants understandings.

Text/Reference
Books,etc. The texts are not fixed. The instructors recommend that you keep some of texts/reference books in which you are interested. Problems for exercises are given as prints made by instructors.

PC or AV used in
Class,etc. Handouts, Microsoft Teams

(More Details) Student presentations will be made using Teams. Exercises and other materials will be distributed via paper and Moodle.

Learning techniques to be incorporated Discussions, PBL (Problem-based Learning)/ TBL (Team-based Learning)

Suggestions on
Preparation and
Review You need to make provide answers for problems offered by instructions. It is very important to prepare your answers even when you do not represent them in the class.

Requirements Participants are strongly recommended to take the course "Analysis IV."

Grading Method Grading is mainly based on the following factors: the results of the quizzes in every time of the lessons, the small examinations of several times, the results of the reports, the numbers of problems which you present your answer in front of all participants. In the beginning of the course, instructors will show the minimum requirements for getting the unit of this course. You should attend the first day of the course.

Practical Experience

Summary of Practical Experience and Class Contents based on it

Message 1. Please refrain from talking in class! If you want to speak, please give us the great answers prepared before the class and demonstrate them for all participant's benefit.
2. In mathematics, it is very clear whether you can solve problems in the course or not. Sometimes such results make you happy or depress you. People want to give right answers and demonstrate them smartly.  But if you can do them from the beginning, you do not take this course. When you start to learn new things, you make many mistakes, but do not be afraid for them. You need to overcome them. Nothing happen even if you give wrong answers in the class. In tomorrow, the sun will raise as is in yesterday. Please make mistakes and find what is the right way in mathematics.

Other In Japanese universities it is not customary to bring food or drinks into the classroom unless the instructor gives permission to do so. No drinking and no eating in the classroom, please!

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	HA095000	Subject Classification	Specialized Education
Subject Name	解析学IV演習
Subject Name （Katakana）	カイセキガク４エンシユウ
Subject Name in English	Exercises in Analysis IV
Instructor	KAWASHITA MISHIO
Instructor (Katakana)	カワシタ　ミシオ
Campus	Higashi-Hiroshima	Semester/Term	2nd-Year, Second Semester, 4Term
Days, Periods, and Classrooms	(4T) Mon1-2,Thur3-4：SCI E209
Lesson Style	Seminar	Lesson Style (More Details)	Face-to-face
Blackboards and projectors are used. Sometime, prints are given.
Credits	1.0	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 students of department of mathematics in faculty of science
Keywords	multiple integral, iterated integral (repeated integral), change of order of integration, aria of sets and measure zero sets in the sense of Jordan, change of variables, improper integral, volume, surface aria
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 is to help you understand the contents of "Analysis IV." The course is given by exercise style.
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 It is possible to change the order of the contents or the contents themselves. Perhaps, some of the quizzes in the class or the reports are given. The proposed plan of the course may change when it is better for participants understandings.
Text/Reference Books,etc.	The texts are not fixed. The instructors recommend that you keep some of texts/reference books in which you are interested. Problems for exercises are given as prints made by instructors.
PC or AV used in Class,etc.	Handouts, Microsoft Teams
(More Details)	Student presentations will be made using Teams. Exercises and other materials will be distributed via paper and Moodle.
Learning techniques to be incorporated	Discussions, PBL (Problem-based Learning)/ TBL (Team-based Learning)
Suggestions on Preparation and Review	You need to make provide answers for problems offered by instructions. It is very important to prepare your answers even when you do not represent them in the class.
Requirements	Participants are strongly recommended to take the course "Analysis IV."
Grading Method	Grading is mainly based on the following factors: the results of the quizzes in every time of the lessons, the small examinations of several times, the results of the reports, the numbers of problems which you present your answer in front of all participants. In the beginning of the course, instructors will show the minimum requirements for getting the unit of this course. You should attend the first day of the course.
Practical Experience
Summary of Practical Experience and Class Contents based on it
Message	1. Please refrain from talking in class! If you want to speak, please give us the great answers prepared before the class and demonstrate them for all participant's benefit. 2. In mathematics, it is very clear whether you can solve problems in the course or not. Sometimes such results make you happy or depress you. People want to give right answers and demonstrate them smartly. But if you can do them from the beginning, you do not take this course. When you start to learn new things, you make many mistakes, but do not be afraid for them. You need to overcome them. Nothing happen even if you give wrong answers in the class. In tomorrow, the sun will raise as is in yesterday. Please make mistakes and find what is the right way in mathematics.
Other	In Japanese universities it is not customary to bring food or drinks into the classroom unless the instructor gives permission to do so. No drinking and no eating in the classroom, please!
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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