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.
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
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course with recitation sessions. Homework will be assigned every week. There will be occasional quizzes.
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
None required
Final exam 50%, assignments and quizzes 50%.
Students are expected to have passed Calculus (I/II), Linear Algebra (I/II), and their recitations.