Introduction to Analysis IV

Academic unit or major

Undergraduate major in Mathematics

Instructor(s)

Hideyuki Miura / Masaharu Tanabe

Class Format

Lecture/Exercise

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Class

-

Course Code

MTH.C204

Number of credits

110

Course offered

2025

Offered quarter

4Q

Syllabus updated

Mar 19, 2025

Language

Japanese

Syllabus

Course overview and goals

This course is a succession of "Introduction to Analysis III" in the third quarter. We will continue to teach "vector calculus", that is a calculus for scalar fields (single-valued functions) and vector fields (multivalued functions) . Each lecture will be followed by a recitation (a problem-solving session).

The students will learn "divergence theorem" and "Stokes' theorem" on surface integrals. They will also learn differential forms to formalize these theorems in a unified manner, as extensions of the "fundamental theorem of calculus".

Course description and aims

At the end of this course, students are expected to:
-- understand the tangent vectors and tangent space of surfaces
-- be able to calculate surface integrals of vector fields
-- understand the meaning of divergence theorem and Stokes' theorem
-- be able to calculate differential forms

Keywords

tangent vector, surface integral, divergence theorem, Stokes theorem,
differential forms, exterior derivative

Competencies

• Specialist skills
• Intercultural skills
• Communication skills
• Critical thinking skills
• Practical and/or problem-solving skills

Class flow

This is a standard lecture course with recitation sessions. Homework will be assigned every week. There will be occasional quizzes.

Course schedule/Objectives

Study advice (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)

None required

Reference books, course materials, etc.

None required

Evaluation methods and criteria

Based on the final exam, quizzes, and the problem solving situation in the recitation sessions. Details will be provided in the class.

Related courses

• MTH.C201 ： Introduction to Analysis I
• MTH.C202 ： Introduction to Analysis II
• MTH.C203 ： Introduction to Analysis III
• MTH.C204 ： Introduction to Analysis IV

Prerequisites

Students are expected to have passed
-- Calculus (I/II), Linear Algebra (I/II), and their recitations.
-- Introduction to Analysis I/II.