In this course we first give a rigorous formalization of "limits" of sequences and functions by means of the "epsilon-delta" definitions. We also learn the differential calculus rigorously, and in particular the Taylor expansion as an excellent polynomial approximation. Next 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. It is strongly recommended to take "Exercises in Analysis A I", which is a complementary recitation session.
Students will learn how to write the mathematical analysis logically. More precisely, the students will become familiar with the "epsilon-delta" definitions and proofs, and be able to describe limits of real numbers rigorously.
At the end of this course, students are expected to:
-- Understand the construction of irrational numbers
-- Be familiar with limits superior and limits inferior
-- Be able to state and prove propositions about limits of sequences and functions by using "epsilons and deltas".
-- Understand some important properties of continuous functions, such as the intermediate value theorem, and the existence of the maximum/minimum.
-- Be able to calculate polynomial approximations of a given function by using the Taylor expansion or asymptotic expansion.
-- 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
Continuity of real numbers,
infimum, supremum,
limit superior, limit inferior,
Cauchy sequence, continuous function,
derivative, Taylor expansion
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. Homework will be assigned every week. There will be occasional quizzes.
Course schedule | Required learning | |
---|---|---|
Class 1 | Continuity of real numbers | Details will be provided in class. |
Class 2 | Limit of series, subsequence, and accumulation point | Details will be provided in class. |
Class 3 | limit superior, limit inferior, Cauchy sequence | Details will be provided in class. |
Class 4 | Convergence of series, interchanging the order of summation | Details will be provided in class. |
Class 5 | Limits and continuity of functions | Details will be provided in class. |
Class 6 | Continuous functions | Details will be provided in class. |
Class 7 | Differentiability, Rolle's theorem, mean-value theorem | Details will be provided in class. |
Class 8 | higher order derivatives and Taylor expansion, quiz | Details will be provided in class. |
Class 9 | Pointwise and uniform convergence of sequences of functions | Details will be provided in class. |
Class 10 | power series | Details will be provided in class. |
Class 11 | Limits and continuity of multivariable functions | Details will be provided in class. |
Class 12 | Total and partial derivative | Details will be provided in class. |
Class 13 | Local maxima and minima of multivariable functions | Details will be provided in class. |
Class 14 | Inverse function theorem and implicit function theorem, method of Lagrange multiplier and its application | Details will be provided in class. |
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
None required
None required
Based on the final exam and quizzes. Details will be provided in the class.
Students are expected to have passed Calculus (I/II), Linear Algebra (I/II), and their recitations.