2022 Faculty Courses School of Computing Undergraduate major in Mathematical and Computing Science

Vector and Functional analysis

Academic unit or major: Undergraduate major in Mathematical and Computing Science
Instructor(s): Hideyuki Miura / Takeshi Gotoda
Class Format: Lecture/Exercise (Face-to-face)
Media-enhanced courses: -
Day of week/Period (Classrooms): 5-6 Tue (W834) / 5-8 Fri (W834)
Class: -
Course Code: MCS.T301
Number of credits: 210
Course offered: 2022
Offered quarter: 3Q
Syllabus updated: Jul 10, 2025
Language: Japanese

Syllabus

Course overview and goals

We will present the vector analysis and the functional analysis as the fundamental tools for the mathematical analysis.
The first half of this course is devoted to the calculus of scalar fields, vector fields.
In the last half of this course the fundamentals of the functional analysis such as Banach spaces, linear operators, Hilbert spaces, orthogonal decompositions and the Riesz representation theorem are given.

Course description and aims

The object of this course is to explain the vector analysis and the functional analysis as the fundamental tools for the mathematical analysis.
By completing this course, students will be able to:
1) understand the integrals of vector fields and master various integral formula.
2) understand fundamental properties of the Banach spaces and linear operators, the orthogonal decomposition and the Riesz representation theorems are given.

Keywords

vector fields, integral formula, Banach space, linear operators, Hilbert space

Competencies

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

Class flow

For the understanding of this course, it is necessary to be skilled at the contents by the calculation by hand. Therefore the exercise class is given.

Course schedule/Objectives

	Course schedule	Objectives
Class 1	Parametrization for curves and surfaces	Understand the contents of the lecture.
Class 2	Gradient, divergence and rotation	Understand the contents of the lecture.
Class 3	Exercise for parametrization for curves and surfaces, and gradient, divergence, rotation	Cultivate a better understanding of lectures.
Class 4	Contour integral and surface integral	Understand the contents of the lecture.
Class 5	Integral theorems	Understand the contents of the lecture.
Class 6	Exercise for contour integral, surface integral and integral theorems	Cultivate a better understanding of lectures.
Class 7	Banach spaces	Understand the contents of the lecture.
Class 8	Contraction mapping principle	Understand the contents of the lecture.
Class 9	Exercise for Banach spaces and contraction mapping principle	Cultivate a better understanding of lectures.
Class 10	Review of Lebesgue's integral	Understand the contents of the lecture.
Class 11	Exercise for Lebesgue's integral	Cultivate a better understanding of lectures.
Class 12	Function spaces	Understand the contents of the lecture.
Class 13	Bounded linear operator	Understand the contents of the lecture.
Class 14	Exercise for function spaces and bounded linear operators	Cultivate a better understanding of lectures.
Class 15	Hilbert spaces	Understand the contents of the lecture.
Class 16	Orthonormal system	Understand the contents of the lecture.
Class 17	Exercise for Hilbert spaces and orthonormal system	Cultivate a better understanding of lectures.
Class 18	Orthogonal decomposition theorem	Understand the contents of the lecture.
Class 19	Riesz representation theorem	Understand the contents of the lecture.
Class 20	Exercise for orthogonal decomposition theorem and Riesz representation theorem	Cultivate a better understanding of lectures.
Class 21	Spectrum theorem	Understand the contents of the lecture.

Study advice (preparation and review)

To enhance effective learning, students are encouraged to spend a certain length of time outside of class on preparation and review (including for assignments), as specified by the Tokyo Institute of Technology Rules on Undergraduate Learning (東京工業大学学修規程) and the Tokyo Institute of Technology Rules on Graduate Learning (東京工業大学大学院学修規程), for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

To be announced

Reference books, course materials, etc.

To be announced

Evaluation methods and criteria

By scores of the examination and the reports.

Related courses

LAS.M101 ： Calculus I / Recitation
LAS.M105 ： Calculus II
LAS.M102 ： Linear Algebra I / Recitation
LAS.M106 ： Linear Algebra II
MCS.T304 ： Lebesgue Interation
MCS.T211 ： Applied Calculus
MCS.T311 ： Applied Theory on Differential Equations

Prerequisites

The students are encouraged to understand the fundamentals in the calculus and the linear algebras.