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## Teaching

See the following webpage for information regarding emergencies:

Beginning in Fall 2019, NetMath will be offering Math 487 (ECE 493). The course will include 32 videos I created, based on my book Linear and Complex Analysis for Applications.

In Spring 2018 I am teaching Math 424 (Honors Real Analysis) and Math 425 (Honors Advanced Analysis)

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Math 424.

Class meets MWF from 10:00 to 10:50 in 343 Altgeld Hall.

Course website address (you are there): https://math.uiuc.edu/~jpda/teaching.html

Text: Introduction to Analysis by Maxwell Rosenlicht. I intend to cover Chapters 1 through 8.

Another plausible text, Elementary Analysis: the theory of calculus'' by Ken Ross, is used in Math 447. It is good, but perhaps a bit too easy for this honors class.

I have written a book called Hermitian Analysis'' that I once used for the capstone honors class Math 428. I may on occasion provide value-added with material from this book.

It might also be useful for you to read chapters 13-17 of Mathematical Thinking (by D'Angelo-West), which is sometimes used for Math 347.

The course will emphasize proofs of the basic theorems of analysis on the real line and in metric spaces.

Course grades will be based upon total points. There will be two quizzes (50 points each), two exams (100 points each), and the final exam (200 points). If needed, we might have an additional quiz.

There will be weekly homework; a few questions on each assignment will be graded. The homework grades will count a small amount in your course grade, the total will be in between a quiz and an exam. In addition, there will be one take home problem set consisting of harder problems. It will be worth 100 points and it will be due April 13.

Quizzes will be given Feb. 9 and Mar. 9. If needed, the third quiz will be given in April.

The exams will be given Feb. 23 and April 6.

The exam on April 6 will cover chapters 4,5,6, emphasizing 5 and 6.

The final exam will be given Wed. May 9, 2018 from 7:00 to 10:00 PM.

Last day of class is Wed., May 2, 2018.

OFFICE HOURS: My office is 355 Altgeld Hall. Office hours will generally be MW 11:00 to 11:45 and MF 2:00 to 3:00, plus other times by appointment.

To make an appointment, ask in class, send me e-mail (jpda@illinois.edu), or call me at 333-6406.

Homework due January 26: Page 12: 3, 5, 7, 8, 9. Page 29: 4a, 7, 9, 10, 11. Do 8 but don't hand it in.

Homework due Feb. 2: Page 61: 2, 3, 6, 10, 11, 12.

Homework due Feb. 9: Page 62: 13, 15, 16, 17, 28, 30, 31, 32, 33.

Homework due Feb 16: Page 63: 18,19, 24, 25 (but don't hand in), 26, hand in 33 again.

Homework due Feb. 23: Study for exam! Hand in Page 63: 35-37. Page 90: 1abcd.

Homework due March 2: Page 90: 3,4, 9ab, 10 bc.

Homework due March 9: Page 90: 17.18,19, 24, 27, 32, 33.

Homework due March 16: Page 90: 34, 36, 42. Page 108: 1,2,3,5,6.

Homework due March 30: Page 132. 2,8,9,10,11. Use Lower and upper sums rather than Riemann sums.

Homework due April 6: Page 132. 16,17,18,21,23,24.

April 13: HAND IN AS MANY OF THE CHALLENGE PROBLEMS below you can do.

Homework due April 20: Page 160. 2, 3 (draw a graph!), 8a, 20, 27, 30.

No Homework due April 27. We will be reviewing at this time.

HOMEWORK set due April 13. I will gradually add problems to this list.

1) Give an example of an ordered field containing the integers but for which the integers are a BOUNDED set. Show in this field that the floor function does not make sense!

2) Show that the function f is infinitely differentiable at every x on R. Here f(x) = e^(-1 / x^2) for x \ne 0 and f(0)=0. (The only issue is at 0.) Construct an infinitely differentiable function that is 0 outside of the interval [a,b] and positive on the open interval (a,b).

3) Give an example of a function on the real line that is differentiable at each integer but nowhere else.

4) Let M be a compact metric space. Let C(M) denote the continuous real-valued functions on M. Define a norm on C(M) by ||f||= sup |f(x)| such that x is in M. Show that C(M) becomes a metric space. Determine the compact subsets of C(M). (Difficult)

5) Prove Riemann's remark. Suppose that \sum_{n=1}^\infty a_n converges conditionally. Show that you can rearrange the terms to make the series sum to any given real number.

6) following up on 5), consider the sequence 1, 1/3, -1/2, 1/5, 1/7, -1/4, 1/9, 1/11, -1/6 ... What is the sum of the series?

7). Do problems 8 and 9 on page 109 in Rosenlicht.

8) and 9) Page 166 numbers 39, 41 in Rosenlicht.

10) Give an example of a sequence a_n such that \sum a_n converges but \sum (a_n)^3 diverges. Justify.

11) Number 30 on page 164 in Rosenlicht. Explain carefully why we can evaluate at x=1.

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Math 425 meets MWF at 12:00 in 345 Altgeld.

Course website address (You are there): https://math.uiuc.edu/~jpda/teaching.html

Text: Calculus on Manifolds by Michael Spivak.

This classic thin short book is wonderful. We will cover the entire book and I will provide considerable value-added.

I have written two books which augment some of the topics in Spivak. Linear and complex analysis for applications'', published by CRC Press, is used for ECE 493 (Advanced Engineering Mathematics). See also Hermitian Analysis'', published by Springer. Drafts of these books appear on my teaching web page.

GRADING: Course grades will be based upon total points. There will be three quizzes (50 points each), one mid-term exam (100 points), and the final exam (200 points). There will be weekly homework; a few questions on each assignment will be graded. The homework grades in total will count about 100 points. There will be a harder homework problem set due April 20 worth 100 points.

PLEASE NOTE CHANGES IN EXAM DATES

EXAM DATES:

Quizzes will be given Feb. 2 and Mar. 5. The third quiz will be given on April 20. It will be computational, involving differential forms, Lie derivatives, Stokes' theorem.

The mid-term exam will be given March 14. NO CLASS on March 16.

The final exam will be given Monday May 7, 2018 from 1:30 to 4:30 PM.

Last day of class is Wed., May 2, 2018.

OFFICE HOURS: My office is 355 Altgeld Hall. Office hours will generally be MW 11:00 to 11:45 and MF 2:00 to 3:00, plus other times by appointment.

To make an appointment, ask in class, send me e-mail (jpda@illinois.edu), or call me at 333-6406.

Homework due January 26: Page 4: 1.2, 1.3, 1.4, 1.7, 1.10, 1.12. Page 10: 1.14, 1.19, 1.21 especially part c. Know both directions of Corollary 1.7 (Exercise 1.20).

Homework due Feb 2: Page 13: 25, 26, 27. Page 18: 2.5, 2.6, 2.7. Page 23: Know how do do 2.10, but don't hand in. 2.11, 2.16.

Homework due Feb. 9. Page 23: 2.14, 2.15. Page 28: 2.24, 2.25. Page 33: 32, 34. Page 39: 37,38. Page 43: 41-c.

Homework due Feb. 16: Page 49: 3.2, 3.5, 3.6. Page 52: 3.12. Page 61: 3.28. 3.33.

Homework due Feb. 23: Page 61: 3.35. Page 66: 3.38. Do also the following computational problems: Suppose f and g are infinitely differentiable in one dimension. First find a general formula for the k-th derivative of f times g. Then, for 1 \le k \le 6, find a formula for the k-th derivative of the composition g \circ f. Then generalize these formulas to higher dimensions.

Homework due March 2: Derive the formula for the volume form in spherical coordinates by using wedge products.

Homework due March 2 (continued).

Let A(x,x) be a quadratic form on R^n. Assume A is positive definite. Find the integral of e^(-A(x,x)) over R^n. (Suggestion: diagonalize the quadratic form to reduce to the one-dimensional case. Then use the standard Gaussian trick of squaring the integral and using polar coordinates)

Homework due March 2 (continued). Use Green's theorem to find the area inside the loop defined by x^3 + y^3 + 3xy. (Comment. parametrize the loop using y=tx.)

Homework due March 9: Page 84: 4.2, 4.9 (just hand in parts d,e). Write down the Hodge * operator on two forms in R^4. (say what *w is for each of six basis elements w). Find the characteristic polynomial of the n by n matrix whose first row is 1,2,3,...n whose second row is n+1,n+2,...2n and whose last row is (n-1)n +1, (n-1)n +2, ..., n^2.

NO HOMEWORK DUE March 16. We have an exam March 14 and no class on March 16.

Homework due March 30: 4.19, 4.20, 4.28 and 4.33 (has many parts).

Homework due April 7. Suppose that f: C^n -> C^n is complex analytic. Let F:R^(2n) -> R^{2n} be the corresponding real variables map. Express det (DF) in terms of Det (f'). Here f' is the complex derivative matrix. (Suggestion: use differenital forms.) Do 5.26 and 5.35.

To prepare for the Quiz April 20, here are some practice problems.

Note that there will be a quiz on April 20.

HOMEWORK SET due April 20.

1) Suppose that X is a subset of a topological space. Consider two operations, closure and complement. By iterating these operations, show that there are at most 14 distinct sets one can get. Give an example of a subset of the real line for which one gets 14 sets.

2) Prove that (\sum_{j=1}^n j x_j)^2 \le (n^3/3 + n^2/2 + n/6) (\sum_{j=1}^n (x_j)^2).

3) Let p(x,y) be a homogeneous polynomial of degree 4. Consider p(x,y)/(x^2 + y^2). Give a necessary and sufficient condition for the second mixed partials p_{xy} and p_{yx} (at the origin) to be different.

4) For (x,y) not (0,0) put f(x,y) = ( |x|^a |y|^b ) / (|x|^(2c) + |y|^(2b)). Here a,b,c,d are positive numbers (not necessarily integers). Give a necessary and sufficient condition on these numbers for f to be continuous at (0,0).

5) Suppose F:R^n -> R^n is a polynomial of degree at most two and det(DF(x)) = 1 for all x. Prove that F is injective. Suggestion. If not, without loss of generality (why?), assume f(p) = f(0)=0 for some p. Consider f(tx) for t in R. Differentiate and choose t to contradict the hypothesis on the Jacobian. Comment: This is an open problem for degree three (or more).

6) Do problem 2.26 page 29 in Spivak.

7) Give an example of a 2-form, say w, on R^(2n) such that w wedge w ... wedge w (n times) is not zero.

8) Use Green's theorem to prove the Cauchy integral formula for complex analytic functions; then prove the more general statement for smooth functions that involves the z-bar derivative if f.

9) Find the volume of the unit ball in R^n.

10) Define a function w by the implicit equation x = w(x) e^(w(x)). Let K'=w. Show that K(x) = x( w(x) + (1/w(x)) - 1). Find the integral of w(x)dx from 0 to e.

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Math 199 Talk Feb. 28, 2018.

Beamer file for this talk. Series:tricks and traps

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The following article discusses the role of complex variables in the curriculum. Please read it.

In Fall 2017 I taught Advanced Engineering Mathematics, cross-listed as Math 487 and ECE 493.

The class met MWF 10 in 241 Altgeld.

Course website: You are there! address: https://math.uiuc.edu/~jpda/teaching.html

Text: Linear and Complex Analysis for Applications, by John P. D'Angelo (CRC Press). (available in both hard-cover and electronic versions)

During the semester we will cover Chapters 1-4, parts of 5,6,7. Interests of the students will determine the specific topics from these later chapters.

The main point of the class will be to understand linear systems in many contexts and to unify our point of view.

Graduate student credit: Graduate students who wish to get 4 credit hours for the course need to get a form from Liz Vonk in the Math Dept. Office in 313 Altgeld Hall. We then agree on what extra work needs to be done. Possibly one or two take-home problem sets, possibly some small project. Details will be discussed the first two days of class.

If the class wishes, we will have one or two diagnostic exams. They don't count, but they will help me decide what backgrounds students have and thereby improve the course for all. Course grades will be based upon total points. There will be two quizzes (50 points each), two exams (100 points each), and the final exam (200 points).

There will be homework; a few questions on each assignment will be graded. The homework grades will count a small amount in your course grade, the total will be in between a quiz and an exam.

He has reserved room AH143 for an office hour each Wednesday from 3:00 -> 4:00 PM.

The quizzes will be given September 15 and October 27.

There will be a practice quiz during the week of Nov. 6.

The exams will be given September 22 and December 1.

The final exam will be Monday December 18, 2018 at 7:00 PM.

Last day of class is Wednesday December 13, 2018.

My office is 355 Altgeld Hall. Office hours will generally be MWF 11:00 to 11:45, W 9:00 to 9:50, plus other times by appointment. To make an appointment, ask in class, send me e-mail (jpda@illinois.edu), or call me at 333-6406.

First Diagnostic Exam: First diagnostic exam

Second Diagnostic Exam: Second diagnostic exam

Residue and Laplace transform practice take home: Residue practice.

STUDENT PRESENTATIONS! Please let me know if and when you wish to give one.

Monday Nov. 6: Ivan Abraham, Ben Steele.

Wednesday Nov. 8: Mei-Yun Lin, Ahmed Alromaithi.

Friday Nov. 10: Aaron Perry, Kyle Pieper.

Wednesday Nov. 15: Ian Grady, Zipeng Wang

Wednesday Dec. 6: Ana Silva, N. Richardson.

Homework: Due Friday Sept. 1: Exercises 1.2, 1.4, 2.2, 2.4, 2.3, 2.7, 3.2, 3.3

Homework: Due Friday Sept. 8: Exercises 2.1, 2.5, 3.5, 3.6, 4.3, 4.5, 4.6, 4.7, 4.8, 5.5

Homework: Due Friday Sept. 15: Exercises 5.6, 5.9, 6.1, 6.2, 6.3, 6.4, 7.1, 7.2, 7.3, 7.4

Homework: Due Friday Sept. 22: (Still from Chapter 1) Exercises 7.7, 7.9, 8.4, 8.5, 9.8, 10.3, 13.2.

Homework: Due Friday Sept. 29: Chapter 2. 1.5, 1.6, 1.8, 2.2, 2.3, 2.9, 3.1, 3.2, 3.6.

Homework: Due Friday Oct. 6: Chapter 2: 3.5, 4.3, 4.7, 5.3, 5.4, 5.6, 5.8, 6.2, 7.4.

Homework: Due Friday Oct. 13: Chapter 3: 1.2 (minimum should be "local minimum") , 1.5, 2.17, 3.1, 3.2, 3.3, 4.1, 4.3, 4.7.

Homework: Due Friday Oct. 20: Chapter 3: 5.2, 5.4, 5.5, 6.2, 6.3, 6.4.

Homework: Due Friday Oct. 27: Chapter 4: 2.1, 2.2, 2.4, 2.5, 3.2.

Homework: Due Friday Nov. 3: Chapter 4: 5.6, 5.7, 6.3,6.4,6.5.

Homework: Due Friday Nov. 10: Chapter 4: 7.1, 7.3, 8.1, 8.4.

Homework: Due Friday Nov. 17: Chapter 4: 8.2, 8.8. Chapter 5: 1.2, 1.3, 1.4 (Use Theorem 1.3 to do 1.4), 1.14.

NOTE: The grader's office hour on Nov. 1 will be held in room 347 Altgeld.

Last Homework. Due Friday Dec. 8: Chapter 5: 1.17, 3.6. Chapter 6: 1.3, 1.4, 4.1, 4.3.

The final will include some basic material on Hilbert spaces.

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In Fall 2016 I taught Math 347 and Math 446.

MATH 347:

Text: Mathematical Thinking: Problem Solving and Proofs, J. D'Angelo and D. West, Prentice-Hall, 2nd edition.

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MATH 446

Some of the lectures are on youtube videos: Here are the links:

https://youtu.be/oxjo6_5qaqA

https://youtu.be/F4hINSjwQgA

https://youtu.be/B0lkqJh0-nM

https://youtu.be/zG8BSKn2cCQ

https://youtu.be/fTskug4FGnw

https://youtu.be/HtIuZonywuE

https://youtu.be/2bAYg-wum-w

https://youtu.be/y5IOiq2Xyl8

https://youtu.be/AZY3KKRS2Uk

https://youtu.be/__Q4w6gaeJU

https://youtu.be/1gsWocDQrCE

https://youtu.be/jRnLETOREbw

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In Spring 2016 I taught a graduate course on Holomorphic Mappings.

In Spring 2016 I also taught Honors Calculus III.

Please click below for the text book, exam dates, and other course information about Honors Calculus III.

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My Spring 2016 Math 595 class discussed holomorphic mappings in several complex variables.

This course began with the basic theory of complex analysis in several variables. We discussed some of the similarities and differences between complex analysis in one and higher dimensions. After proving some of the main facts about the Cauchy-Riemann equations and pseudoconvex domains, including solving d-bar, we considered holomorphic mappings and CR Geometry. We mentioned work about CR mappings between spheres in different dimensions, allowing us to see connections with harmonic analysis, representation theory, algebraic combinatorics, and other parts of mathematics.

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The following article discusses the role of complex variables in the curriculum. Please read it.

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In Fall 2015 I taught Math 446, Applied Complex Variables.

Spring 2015 I taught Math 428.

In Fall 2014 I taught (the Honors section of) Abstract Linear Algebra, Math 416, MWF at 9:00.

In Fall 2013 I taught Math 198 and Math 416 Honors.

In Spring 2014 I taught Advanced Engineering Mathematics (Math 487, ECE 493).

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