Math 430/431 Mathematical Analysis (Some call
this Real Analysis or Advanced Calculus.). The on-screen textbook can be downloaded from this link.
- Course Contract
- We will cover some topics from chapters 3 and 4 for review. The main content for 430 starts from chapter 5.
- Proving 1=2 (what went wrong?)
- Proving All People in Canada are the Same Age (what went wrong? (Need principle of induction)
- Interactive Real Analysis.
- Maple command to mean value theorem.
- Page 46-47 (Maple file)
- Homework: page 25, page 28, page 31.
- A problem from page 65 (Maple file).
- Mathematical Induction.
- Countable and uncountable sets (1)
- Page68#17(b).mws
- (0,1) is uncountable. (there is a typo in this page!)
- Definition of field/ring.
- Solving inequalities graphically. (page 83).
- Explore the set of rational number is dense in R. (a Maple file).
- limit points and closed set.
- Hints for problems on Final.
- Hints to a homework.
- Hint on an old Test 1.
- Using Maple to learn sequences.
- About Recursive Sequence.
- Using Fixed Point or
Newton's method? When will 2^x>x^10? Corresponding Maple file
- Another Recursive Sequence with Maple.
- Newton's Method
- A tutorial. (html file)
- Newton's Method (A Maple file)
- **Use Newton's Method to find the inflection point of a function. (Maple file)
- Homework page 150
- Homework page 155
- Cauchy Sequence
- The speed of convergence of two series. (Maple file)
- A link to an online Real Analysis course.
- Using Maple to explore the limit of a function at point. (Maple file).
- Epsilon-delta concept.
- A ruler function
- Taylor polynomial, Fourier Series and Bernstein Polynomial.
- Another look at exploring the limit of a function at point. (Maple file).
- A proof to the squeezing principle.
- Homework set 1 (Exercises on Cantor Theorem)
- Recall the relationship between a closed set and its limit points.
- Solution to page 175.
- Solution to page 195
- Understand the proofs of the followings:
- A continuous function sends a closed and bounded set to a closed and bounded set.
- If f is continuous on a closed and bounded set, then f assumes its maximum and minimum.
- If f is continuous on a closed and bounded set, then f assumes all its intermediate value.
- Solution to (continuous functions on closed and bounded set).
- Solution to page 216.
- Some exercises on uniform continuous functions.
- About continuity and uniform continuity of a function.
- More about uniform continuity
- Continuity and Differentiability
- *A nowhere differentiable function
- A PDF file
- A Maple file.
- Mean Value and Cauchy Mean Value Theorems (Dr. Yang's eJMT paper. A video clip for the proof of MVT.
- Converse of Mean Value Theorem (Dr. Yang's).
- Java applet on Mean Value Theorem.
- Cauchy Mean Value Theorem and L'Hopital's Rule
- Solution to page 237
- Taylor's Theorem.
- Dr. Yang's eJMT paper regarding Mean-Value and Cauchy Mean Value Theorems.
- A Maple template for take home test 2, problem 5.
- Power Series and Taylor's Series-local but not global. (Maple file)
- Ratio test and interval of convergence (html).
- Reading materials (radius of convergence and etc.)
- Second Partial Derivative Test (html).
- Fourier Series approximation is global but not local (Maple).
- Homework on Taylor Polynomial and its Remainder. (PDF)
- Maple solution.
- Motivation for Riemann Integration
- Riemann Integration Theory
- Uneven partition and numerical integrations with singularities
- Lecture 1. (PDF file)
- Lecture 2 (PDF file)
- eJMT Oct 2009 issue (by Yang, Lee and Ding).
- Fundamental Theorem of Calculus and computational HK-integration (by Yang and Ding at the ATCM 2010 Proceedings).
- My own adaptive quadratures, good for functions that are monotone with singularities or highly oscillatory.
- Fubini's Theorem
- *Animations on sequence of functions (Maple)
- Romberg Integration
- About Fubini's Theorem 1
- About Fubini's Theorem, double integral and etc.
- Animations for numerical integration
- Numerical Method.
- Animation on sequence of functions. (Maple file)
- Fourier Series approximation is global but not local (Maple).
- Introduction to Topology.
- Hilbert space and Banach space.
- Cauchy Completeness and Hilbert space and Banach space.
- Online Mathematical Analysis
- Online Multivariable Calculus