2019 Introduction to Analysis IV

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Kagei Yoshiyuki  Miura Tatsuya 
Course component(s)
Lecture / Exercise
Day/Period(Room No.)
Mon3-8(H103)  
Group
-
Course number
MTH.C204
Credits
2
Academic year
2019
Offered quarter
4Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

This course is a succession of "Introduction to Analysis III" in the third quarter. We will continue to teach "vector calculus", that is a calculus for scalar fields (single-valued functions) and vector fields (multivalued functions) . Each lecture will be followed by a recitation (a problem-solving session).


The students will learn "divergence theorem" and "Stokes' theorem" on surface integrals. They will also learn differential forms to formalize these theorems in a unified manner, as extensions of the "fundamental theorem of calculus".

Student learning outcomes

At the end of this course, students are expected to:
-- understand the tangent vectors and tangent space of surfaces
-- be able to calculate surface integrals of vector fields
-- understand the meaning of divergence theorem and Stokes' theorem
-- be able to calculate differential forms

Keywords

tangent vector, surface integral, divergence theorem, Stokes theorem,
differential forms, exterior derivative

Competencies that will be developed

Intercultural skills Communication skills Specialist 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 Parametrization of surfaces and tangent spaces Details will be provided in class.
Class 2 Recitation Details will be provided in class.
Class 3 Surface area and surface integrals Details will be provided in class.
Class 4 Recitation Details will be provided in class.
Class 5 Gauss' divergence theorem Details will be provided in class.
Class 6 Recitation Details will be provided in class.
Class 7 Stokes' theorem Details will be provided in class.
Class 8 Recitation Details will be provided in class.
Class 9 Applications of divergence and Stokes' theorems Details will be provided in class.
Class 10 Recitation Details will be provided in class.
Class 11 Poisson's equation Details will be provided in class.
Class 12 Recitation Details will be provided in class.
Class 13 Differentia forms, wedge product, exterior derivative Details will be provided in class.
Class 14 Recitation Details will be provided in class.
Class 15 Integration of differential forms and generalized Stokes' theorem, quiz Details will be provided in class.

Textbook(s)

None required

Reference books, course materials, etc.

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.C202 : Introduction to Analysis 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.

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