### 2016　Vector and Functional analysis

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Undergraduate major in Mathematical and Computing Science
Instructor(s)
Nishibata Shinya  Murata Miho  Miura Hideyuki
Course component(s)
Lecture / Exercise
Day/Period(Room No.)
Mon5-6(W834)  Thr5-8(W834)
Group
-
Course number
MCS.T301
Credits
3
2016
Offered quarter
1Q
Syllabus updated
2016/4/27
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 every two weeks.

### Course schedule/Required learning

Course schedule Required learning
Class 1 exterior product Understand the contents of the lecture.
Class 2 scalar fields, vector fields Understand the contents of the lecture.
Class 3 exerciser of exterior product, scalar fields and vector fields Cultivate a better understanding of lectures.
Class 4 parametrization for curves and surfaces Understand the contents of the lecture.
Class 5 gradient of scalar fields Understand the contents of the lecture.
Class 6 Exercise of parametrization for curves and surfaces, and gradient of scalar fields Cultivate a better understanding of lectures.
Class 7 divergence and rotation of vector fields Understand the contents of the lecture.
Class 8 contour integral, surface integral Understand the contents of the lecture.
Class 9 Exercise of divergence, rotation and integrals Cultivate a better understanding of lectures.
Class 10 Green's formula, Gauss' divergence theorem Understand the contents of the lecture.
Class 11 Stokes' theorem Understand the contents of the lecture.
Class 12 Exercise of integral formula Cultivate a better understanding of lectures.
Class 13 normed space Understand the contents of the lecture.
Class 14 Banach space Understand the contents of the lecture.
Class 15 exercise for Cultivate a better understanding of lectures.
Class 16 bounded linear operator Understand the contents of the lecture.
Class 17 Hilbert spaces Understand the contents of the lecture.
Class 18 exercise for bounded linear operators and Hilbert spaces Cultivate a better understanding of lectures.
Class 19 orthonormal system Understand the contents of the lecture.
Class 20 orthogonal decomposition theorem Understand the contents of the lecture.
Class 21 exercise for orthonormal system and orthogonal decomposition theorem Cultivate a better understanding of lectures.
Class 22 Riesz representation theorem Understand the contents of the lecture.

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

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

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