### 2022　Vector and Functional analysis

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Undergraduate major in Mathematical and Computing Science
Instructor(s)
Miura Hideyuki  Gotoda Takeshi
Class Format
Lecture / Exercise    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Tue5-6(W834)  Fri5-8(W834)
Group
-
Course number
MCS.T301
Credits
3
2022
Offered quarter
3Q
Syllabus updated
2022/9/5
Lecture notes updated
-
Language used
Japanese
Access Index ### Course description and aims

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.

### Student learning outcomes

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 that will be developed

 ✔ 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/Required learning

Course schedule Required learning
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.

### Out-of-Class Study Time (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.

To be announced

To be announced

### Assessment criteria and methods

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 (i.e., required knowledge, skills, courses, etc.)

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