2019 Advanced Calculus II

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Academic unit or major
Mathematics
Instructor(s)
Kagei Yoshiyuki 
Course component(s)
Lecture
Day/Period(Room No.)
Mon3-4(H103)  
Group
-
Course number
ZUA.C203
Credits
2
Academic year
2019
Offered quarter
3-4Q
Syllabus updated
2019/3/18
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

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 Curves and surfaces in the space Details will be provided in class.
Class 2 Curves and surfaces in the space Details will be provided in class.
Class 3 scalar fields and gradient vectors Details will be provided in class.
Class 4 Line integrals of vector fields Details will be provided in class.
Class 5 Green's theorem and its application, 1 Details will be provided in class.
Class 6 Green's theorem and its application, 2 Details will be provided in class.
Class 7 Surface integrals and divergence theorem Details will be provided in class.
Class 8 surface integrals and divergence theorem Details will be provided in class.
Class 9 Parametrization of surfaces and tangent spaces Details will be provided in class.
Class 10 Surface area and surface integrals Details will be provided in class.
Class 11 Gauss' divergence theorem Details will be provided in class.
Class 12 Stokes' theorem Details will be provided in class.
Class 13 Applications of divergence and Stokes' theorems Details will be provided in class.
Class 14 Differential forms, wedge product, exterior derivative Details will be provided in class.
Class 15 Integration of differential forms and generalized Stokes' theorem, comprehension check-up 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 and quizzes. Details will be provided in the class.

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.

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