HA080000 Analysis IV

Hiroshima University Syllabus

Japanese

Academic Year 2024Year 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 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 E210

Lesson Style Lecture Lesson Style
(More Details) 　

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

Credits 2.0 Class Hours/Week 　 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 maxima, minima, extrema, Lagrange multiplier, 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, Gauss's theorem (divergence theorem)

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 the continuation of Analysis I, II and III for freshmen. The contents consist of one of the fundamental subjects which are very important to study mathematics at university although they are still standard calculus. The last thing is “multiple integration”, which is the main target in this course.

Class Schedule lesson1 maxima and minima of function of several real variables
lesson2 constrained maxima and minima
lesson3 Multiple integrals (No.1) (definitions of multiple integrals on rectangles and fundamental properties)
lesson4 Multiple integrals (No.2) (A continuation of fundamental properties, and integrable functions)
lesson5 Multiple integrals (No.3) (iterated integrals and multiple integrals for bounded sets)
lesson6 Multiple integrals (No.4) (definition of multiple integrals and measure zero sets in the sense of Jordan)
lesson7 Multiple integrals (No.5) (measure zero sets and measureable sets in the sense of Jordan)
lesson8 The midterm examination
lesson9 Area (double integrals) and volume (triple integrals)
lesson10 Improper integrals (No.1) (for non-negative functions)
lesson11 Improper integrals (No.2) (for any function, differences from one dimensional case)
lesson12 Multiple integrals (No.6) (the meanings of Jacobian of diffeomorphism)
lesson13 Multiple integrals (No.7) (change of variables)
lesson14 Multiple integrals and how to change procedures taking limits
lesson15 Lengths and surface arias (recall of lengths, and definition of surface aria)

Around the midterm of the course, the midterm examination is held. This may not be the plan exactly. The reports are sometime imposed. Some of the reports may be for preparations of the following lessons.

The proposed plan of the course may change according to situations of the class. Please accept changes times when lecture and exercise are done if the instructors have to do that.

Text/Reference
Books,etc. For students accessible in Japanese, the textbook is almost fixed. Please see Japanese version. If you feel hard for reading by such a minor language, please contact the instructor. Basically any book entitled "calculus" is good for your textbook if you are interested in it.  But, the text should contain so called epsilon-delta arguments.

PC or AV used in
Class,etc. 　

(More Details) Blackboards and projectors are usually used. Copied documents are also used.

Learning techniques to be incorporated 　

Suggestions on
Preparation and
Review You need to make preparations and revisions for each lessons. Particularly, reviewing process after the lessons is important. I hope that you catch this truth from the experience in last year. To understand the contents of the lessons, you need to consider what you have to do, of what you take care, what you should do after the lessons. This is common remarks for all lessons through the first to the last. Self-education is required for people living in the future.

Requirements The course "Exercises in Analysis IV" is given for helping and deepening your understanding. It is strongly recommended that you take the course. Please attend "Exercises in Analysis IV" and understand calculus deeply via solving problems for the exercise.

Grading Method Grading is mainly based on the results of the midterm and the end of term examinations. The rates of them are given as X% for the midterm examination and Y% for the end of the term examination, where $40 \leq X, Y \leq 60$, $X+Y = 100$. These rates depend on the participants. Even if you do not get a good $X$ score, you do not need to give up. In any case, do your best for the last examination. The imposed reports as homework are not into consideration for estimations of the grade, however, the persons who do not submit these tasks too much are considered as giving up getting the units for this 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 lectures instead of the instructor.
2. The mathematical classes given for sophomore make fundamental parts for forthcoming studies in the university.  Your university life will be changed if you can catch the contents of these classes. The contents of the courses gradually turns to specialized as the mathematical courses. You need to grasp these changes well.
3. The speed of the classes will turn to be faster than the ones for freshmen. Gradually, you can not handle studies in university only using knowledge stored in high schools. Please remind yourself why you wanted to go to university and for what you are waiting in university. The instructor is waiting for you at the class.

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	2024Year	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	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 E210
Lesson Style	Lecture	Lesson Style (More Details)
Blackboards and projectors are used. Sometime, prints are given.
Credits	2.0	Class Hours/Week		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	maxima, minima, extrema, Lagrange multiplier, 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, Gauss's theorem (divergence theorem)
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 the continuation of Analysis I, II and III for freshmen. The contents consist of one of the fundamental subjects which are very important to study mathematics at university although they are still standard calculus. The last thing is “multiple integration”, which is the main target in this course.
Class Schedule	lesson1 maxima and minima of function of several real variables lesson2 constrained maxima and minima lesson3 Multiple integrals (No.1) (definitions of multiple integrals on rectangles and fundamental properties) lesson4 Multiple integrals (No.2) (A continuation of fundamental properties, and integrable functions) lesson5 Multiple integrals (No.3) (iterated integrals and multiple integrals for bounded sets) lesson6 Multiple integrals (No.4) (definition of multiple integrals and measure zero sets in the sense of Jordan) lesson7 Multiple integrals (No.5) (measure zero sets and measureable sets in the sense of Jordan) lesson8 The midterm examination lesson9 Area (double integrals) and volume (triple integrals) lesson10 Improper integrals (No.1) (for non-negative functions) lesson11 Improper integrals (No.2) (for any function, differences from one dimensional case) lesson12 Multiple integrals (No.6) (the meanings of Jacobian of diffeomorphism) lesson13 Multiple integrals (No.7) (change of variables) lesson14 Multiple integrals and how to change procedures taking limits lesson15 Lengths and surface arias (recall of lengths, and definition of surface aria) Around the midterm of the course, the midterm examination is held. This may not be the plan exactly. The reports are sometime imposed. Some of the reports may be for preparations of the following lessons. The proposed plan of the course may change according to situations of the class. Please accept changes times when lecture and exercise are done if the instructors have to do that.
Text/Reference Books,etc.	For students accessible in Japanese, the textbook is almost fixed. Please see Japanese version. If you feel hard for reading by such a minor language, please contact the instructor. Basically any book entitled "calculus" is good for your textbook if you are interested in it. But, the text should contain so called epsilon-delta arguments.
PC or AV used in Class,etc.
(More Details)	Blackboards and projectors are usually used. Copied documents are also used.
Learning techniques to be incorporated
Suggestions on Preparation and Review	You need to make preparations and revisions for each lessons. Particularly, reviewing process after the lessons is important. I hope that you catch this truth from the experience in last year. To understand the contents of the lessons, you need to consider what you have to do, of what you take care, what you should do after the lessons. This is common remarks for all lessons through the first to the last. Self-education is required for people living in the future.
Requirements	The course "Exercises in Analysis IV" is given for helping and deepening your understanding. It is strongly recommended that you take the course. Please attend "Exercises in Analysis IV" and understand calculus deeply via solving problems for the exercise.
Grading Method	Grading is mainly based on the results of the midterm and the end of term examinations. The rates of them are given as X% for the midterm examination and Y% for the end of the term examination, where $40 \leq X, Y \leq 60$, $X+Y = 100$. These rates depend on the participants. Even if you do not get a good $X$ score, you do not need to give up. In any case, do your best for the last examination. The imposed reports as homework are not into consideration for estimations of the grade, however, the persons who do not submit these tasks too much are considered as giving up getting the units for this 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 lectures instead of the instructor. 2. The mathematical classes given for sophomore make fundamental parts for forthcoming studies in the university. Your university life will be changed if you can catch the contents of these classes. The contents of the courses gradually turns to specialized as the mathematical courses. You need to grasp these changes well. 3. The speed of the classes will turn to be faster than the ones for freshmen. Gradually, you can not handle studies in university only using knowledge stored in high schools. Please remind yourself why you wanted to go to university and for what you are waiting in university. The instructor is waiting for you at the class.
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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