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- Academic unit or major
- Mathematics
- Instructor(s)
- Kawahira Tomoki
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Mon3-4(H103)
- Group
- -
- Course number
- ZUA.C203
- Credits
- 2
- Academic year
- 2017
- Offered quarter
- 3-4Q
- Syllabus updated
- 2017/3/17
- 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 | Curves and surfaces in the space | Details will be provided in class. |
| Class 2 | Curves and surfaces in the space | |
| Class 3 | scalar fields and gradient vectors | |
| Class 4 | Line integrals of vector fields | |
| Class 5 | Green's theorem and its application, 1 | |
| Class 6 | Green's theorem and its application, 2 | |
| Class 7 | Surface integrals and divergence theorem | |
| Class 8 | surface integrals and divergence theorem | |
| Class 9 | Parametrization of surfaces and tangent spaces | |
| Class 10 | Surface area and surface integrals | |
| Class 11 | Gauss' divergence theorem | |
| Class 12 | Stokes' theorem | |
| Class 13 | Applications of divergence and Stokes' theorems | |
| Class 14 | Differential forms, wedge product, exterior derivative | |
| Class 15 | Integration of differential forms and generalized Stokes' theorem, comprehension check-up |
Textbook(s)
None required
Reference books, course materials, etc.
None required
Assessment criteria and methods
Final exam 50%, assignments and quizzes 50%.
Related courses
- ZUA.C204 ： Exercises in Analysis A II
- 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.
-- Introduction to Analysis I/II.
