2017 Introduction to Analysis II

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Academic unit or major
Undergraduate major in Mathematics
Kawahira Tomoki  Shibata Masataka  Tanabe Masaharu 
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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


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
Class 3 Limit of sequence of functions, interchange of differentiation and integration
Class 4 Recitation
Class 5 Power series
Class 6 Recitation
Class 7 Limits and continuity of multivariable functions
Class 8 Recitation
Class 9 Total and partial derivatives
Class 10 Recitation
Class 11 Local maxima and minima of multivariable functions
Class 12 Recitation
Class 13 Inverse function theorem and implicit function theorem
Class 14 Recitation
Class 15 Method of Lagrange multiplier and its application, quiz


None required

Reference books, course materials, etc.

None required

Assessment criteria and methods

Final exam 50%, assignments and quizzes 50%.

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.

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