### 2021　Introduction to Analysis II

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Kagei Yoshiyuki  Miura Tatsuya
Course component(s)
Lecture / Exercise    (対面/ZOOM)
Day/Period(Room No.)
Mon3-4(H103)  Mon5-8(H103)
Group
-
Course number
MTH.C202
Credits
2
Academic year
2021
Offered quarter
2Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

In this course we give a rigorous formalization of "limits" of sequences functions, "limits" of multivariable functions, and their derivatives by means of the "epsilon-delta" definitions. We also learn how to find local maxima and minima of a given multivariable function. Each lecture will be followed by a recitation (a problem-solving session). This course is a succession of "Introduction to Analysis I" in the first quarter.

The students will learn how to write the multivariable analysis logically. More precisely, the students will become familiar with the "epsilon-delta" definitions and proofs, and be able to describe multivariable calculus rigorously.

### Student learning outcomes

At the end of this course, students are expected to:
-- Understand the difference between pointwise and uniform convergences
-- Be familiar with calculus of power series in the disk of convergence
-- Understand the differentiability of multivariable functions as linear approximations
-- Understand the relation between gradient vectors and partial derivatives
-- Be able to calculate partial derivatives of composed functions
-- Understand the principle of the method of Lagrange multiplier

### Keywords

Uniform convergence, power series, total derivative, partial derivative, Taylor expansion of multivariable functions, inverse function theorem, implicit function theorem, the method of Lagrange multiplier

### Competencies that will be developed

 ✔ 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/Required learning

Course schedule Required learning
Class 1 Pointwise and uniform convergence of sequences of functions Details will be provided in class.
Class 2 Recitation Details will be provided in class.
Class 3 Limit of sequence of functions, interchange of differentiation and integration Details will be provided in class.
Class 4 Recitation Details will be provided in class.
Class 5 Power series Details will be provided in class.
Class 6 Recitation Details will be provided in class.
Class 7 Limits and continuity of multivariable functions Details will be provided in class.
Class 8 Recitation Details will be provided in class.
Class 9 Total and partial derivatives Details will be provided in class.
Class 10 Recitation Details will be provided in class.
Class 11 Local maxima and minima of multivariable functions Details will be provided in class.
Class 12 Recitation Details will be provided in class.
Class 13 Inverse function theorem and implicit function theorem. Method of Lagrange multiplier and its application Details will be provided in class.
Class 14 Recitation Details will be provided in class.

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

None required

None required

### Assessment criteria and methods

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.C203 ： Introduction to Analysis III
• MTH.C204 ： Introduction to Analysis IV

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

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