Professor Wei-Chi Yang

Advanced Calculus (Math 430 & 431)  

Math 430
Before you are qualified to take this case, you need to solve the donkey problem. And you need to see the followings:

• Implicit Differentiation (A flash)
• Derivative of a polar equation (A flash)

1. Course Contract
2. Officially, we will cover from Chapter 5. You need to preview chapters 3 and 4 on your own.
3. NOTES ON SYMBOLIC LOGIC
4. A link to an online advanced calculus course.
5. Proving 1=2 (what went wrong?)
6. Proving All People in Canada are the Same Age (what went wrong? (Need principle of induction)
7. Useful information for advanced calculus.
8. Interactive Real Analysis.
9. Maple command to mean value theorem.
10. Page 46-47 (Maple file)
11. Homework: page 25, page 28, page 31.
    • Negation of a statement involving variables.
    • Picturing negation with quantifiers
12. A problem from page 65 (Maple file).
13. Homework: pages 41, 43, 48
14. Mathematical Induction.
    • Picture proof of sigma i
    • Some examples.
15. Countable and uncountable sets (1)
16. Page 58; pages 66-68
17. Page68#17(b).mws 
18. (0,1) is uncountable. (there is a typo in this page!)
19. Definition of field/ring.
20. Definition of Upper Bound and its applications.
21. The supremum of a set is either in the set or a limit point
    • Proving sup(A+B)=sup(A)+sup(B).
22. Solving inequalities graphically. (page 83).
23. homework page 83
24. homework page 87
25. homework page 89
26. homework page 96
27. homework page 97
28. Explore the set of rational number is dense in R. (a Maple file). 
29. homework page 105
30. homework page 110
31. **homework page 120.
32. homework on limit points.
33. limit points and closed set.
34. (December 1) Using Maple to learn sequences.
35. (December 1) About Recursive Sequence. (PDF file)
    • A Maple file.
36. (December 1) Using Fixed Point or Newton's method?
    • Another Recursive Sequence with Maple.
37. More about Fixed Point and Newton's methods.
38. Newton's Method
    • A tutorial. (html file)
    • Newton's Method (A Maple file)-Lab 1.
    • **Use Newton's Method to find the inflection point of a function. (Maple file)
39. Hints to a homework.
40. homework page 142.

Math 431

1. Homework page 150
2. Homework page 155
   • How does the number e come about?
   • Euler number and a sequence.
3. Cauchy Sequence
   • It is NOT enough to say that if the difference of successive terms is small, then the sequence is Cauchy.
4. The speed of convergence of two series. (Maple file)
5. A link to an online Real Analysis course.
6. Using Maple to explore the limit of a function at  point. (Maple file). 
7. Epsilon-delta concept.
   • Drill type of finding limit using epsilon and delta.
8. A ruler function
9. Taylor polynomial, Fourier Series and Bernstein Polynomial.
10. Another look at exploring the limit of a function at  point. (Maple file). 
11. A proof to the squeezing principle. 
12. Homework set 1 (Exercises on Cantor Theorem)
13. Recall the relationship between a closed set and its limit points.
14. Solution to page 175.
15. Solution to page 195
16. 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.
17. Solution to (continuous functions on closed and bounded set).
18. Solution to page 216.
19. Some exercises on uniform continuous functions.
20. **More about continuity and uniform continuity of a function.
21. **Continuity and Differentiability.
22. A nowhere differentiable function
    • A PDF file
    • A Maple file.
23. Converse of Mean Value Theorem (Dr. Yang's).
24. Cauchy Mean Value Theorem (Dr. Yang's).
25. Java applet on Mean Value Theorem.
26. Cauchy Mean Value Theorem and L'Hopital's Rule
27. Solution to page 237
28. Taylor's Theorem.
    • Power Series and Taylor's Series (Maple file)
    • Taylor's series approximations.
    • More on errors.
    • Reading materials (radius of convergence and etc.)
29. Homework on page 303
30. Uneven partition and numerical integrations with singularities
    • Lecture 1. (PDF file)
    • Lecture 2 (PDF file)
31. My own adaptive quadratures, good for functions that are monotone with singularities or highly oscillatory.
    • 1 dimensional closed quadrature (Maple, Matlab-updated on April 28, 2006)
      • f(x)=1/sqrt(1-x^2) in [-1,0] Maple
    • 1 dimensional open quadrature (Maple, Matlab-updated on April 28, 2006)
    • 2 dimensional closed quadrature (Maple, Matlab-updated on April 28, 2006)
    • 2 dimensional open quadrature (Maple)
32. Romberg Integration
33. About Fubini's Theorem 1
34. About Fubini's Theorem, double integral and etc.
35. Animations for numerical integration
36. Numerical Method.
37. Introduction to Topology.
38. Hilbert space and Banach space.
39. Cauchy Completeness and Hilbert space and Banach space.
40. On-line Multivariable Calculus