Math 382, Advanced Calculus, Spring 2013
This website updated Feb. 11, 2013 (the URL uses M381)
Recent additions and changes will usually be above the first horizontal line. Course policies are here.


Due Wednesday, Feb. 20: Section 8.1, page 340:  1b,e,g,h   [These are more interesting than they first appear. We will be taking limits of integrals and limits of derivatives and limits of continuous functions and not always getting what you might expect. Examples like these are relevant]

Click here for a previous exam on Chapter 7.

Friday, Feb. 22:  No HW due. Exam on Chapter 7.

Due Monday, Feb. 25:  (8.1) 2b,c,d,e,g,h,j

Later (dates to be announced): Section 8.2, page 346: 3, 4, 5, 8a
                                                Section 8.3, page 350:  2 (Do not use 8.3.4), 3, 5 (for part c, you may consider the decreasing case instead if you prefer. I find it more natural to consider decreasing to the zero function.)  [Look at 8. This is important and discussed at length in measure theory, in Math 547, Real Analysis.] 

Chapter 8 comments:  All results for sequences will be converted to results for series. The sequence results then refer to the sequence of partial sums.
Example 8.1.3 is major (and illustrated on page 335).
Example 8.1.6 is an important counterexample.
Definition 8.2.1 of uniform convergence is a key to the chapter.
Example 8.2.4 (page 342f), pictured on page 344 at the top, is a major example.
Theorem 8.2.8 says there is a Cauchy Criterion for uniform convergence. We will use it.
Theorem 8.3.1 is important:  A sequence of continuous functions that converges uniformly converges to a continuous limit.
    There is a corollary for series.  (Theorem 8.4.3)
    There is a way to prove series are uniformly convergent (Theorem 8.4.11 -- the Weierstrass M-test)
Theorem 8.3.3 is important:  The limit of the integral is the integral of the limit, if the convergence is uniform and the functions are continuous.
    There is a corollary for series (Theorem 8.4.15, page 355)
Theorem 8.3.4 is a refinement that we will not use.
Theorem 8.3.5 is worthy of study. When is the derivative of the limit the limit of the derivatives?
        There is a corollary for series.  (Theorems 8.4.17 and 18)
Theorem 8.3.6 makes 8.3.5 slightly more convenient to actually use.  It is, however, missing a hypothesis. We need the derivatives integrable.
        There is a corollary for series.
Theorem 8.3.7 is a good refinement that we will not have occasion to use.
There is a way to prove series are uniformly convergent (Theorem 8.4.11 -- the Weierstrass M-test)
The M-test is a sledgehammer that works most of the time. If you need a more-subtle test, there are some, but we will not use them (Theorems 8.4.13 and 14).
The integral of a sum is the sum of the integrals, if the conditions are right (Theorem 8.4.15)
The derivative of a sum is the sum of the derivatives, if the conditions are right (Theorems 8.4.17 and 18).

Feb. 7
Due Friday, Feb. 8:  Section 7.2, page 309: [show the test] 1a, c, h, j, n, 2

HW for Monday, Feb. 11:   The handout you got in class is here. Feel free to stop after two hours. Please make some conjectures (easy ones) and resolve them.

HW due Wednesday, Feb. 13:  Section 7.2, page 310, 5, 7, Section 7.3, page 315: 2, 7a, 9a,b,c 

Due Friday, Feb. 15:  Section 7.4, page 322: 2a, 3a, 5 (3 series, one for each part), 7, 8.  Memorize the proof of the alternating series test (T7.4.2).  Section 7.1, #16, using tools from Sections 7.2-7.4.

Due Monday, Feb. 18: Section 7.3, page 317, 11, 12.  Section 7.5 [New rules: If it is true, just say so. If it is false, give a counterexample]  4, 5, 6, 9, 12, 16, 18, 19

Jan. 30
NBC News says math and stat are good careers:
http://careers.msnbc.msn.com-iw3.net/finance/?dBcMxg

Jan. 23
A previous exam on Chapter 6.

Friday, Jan. 25: Prove our two class claims:  Claim 1: If the integral of g2 is 0, then the integral of the absolute value of g is zero.
Claim 2:  If f is Riemann integrable, then f 2 is Riemann integrable.
Note: We reduced the second to proving Claim 3:  If f is Riemann integrable, then
sup(f 2) - inf(f 2) is at most some constant times sup(f) - inf(f).
Limitation:  Spend at most two hours on this. Hand in something.

 [Postponed till Monday] We had two versions of the Fundamental Theorem of Calculus: 6.4.2 and 6.4.4b. Memorize the proof of 6.4.2 and a proof (maybe a variant of the given one) of 6.4.2b. After all, it is the Fundamental Theorem of Calculus. Hand in: Section 6.4,  page 263:  10f, 20 (a neat, non-trivial, problem confirming work of Archimedes who died in the third century BC).    
    Note the existence of problems 6.4: 14 and 16 about Taylor's Theorem. Don't do them.

The previously announced HW has been moved one class day

Due Wednesday, Jan. 30: Section 6.5:  page 270:  2, 3, 9a, b  [Note: Definitions 6.5.1 and 6.5.4 are the key parts of Section 6.5. We will not emphasize 6.5.16-18, but 6.5.19 is interesting.]
                     Section 6.7 (Instructions:  Say if it is true [no proof required], and if it is false, give a counterexample)  page 283: 2, 3, 4, 7, 8

Due Friday, Feb. 1:  Section 7.1, page 300: 1a, c (this is telescoping; find the sum), 3a,c, 4, 5, 9 [We will do #10 with the letters a and b interchanged, in class]

Monday, Feb. 4: Exam on Chapter 6. No HW due,

Due Wednesday, Feb. 7:  Section 7.1, page 302, 20. Also, read Theorem 7.2.1 and its proof and determine the role of the hypothesis "continuous."

Comments:  Only two types of series are easy to sum explicitly, geometric series (page 297, Theorem 7.1.14, which is very important) and telescoping series (which will be covered more thoroughly in class than he does. He barely mentions it at Example 7.1.7, page 296.)
Example 7.1.13 is very important. We can prove it diverges two distinct ways.
Theorems 7.1.8-9 are just series versions of previous sequence results.
Corollary 7.1.21 is important (worthy of being called a theorem), and its converse is false.
Theorem 7.2.1 is important and has a good picture. Example 7.2.3, p-series, is important.
7.2.7, the Limit Comparison Test, is important.
We can get along without Theorem 7.3.1. 7.3.3 is basically a comparison with the geometric, and can be proven that way without 7.3.1.
7.3.5b is obvious, since the terms don't go to zero.  7.3.5a is a Corollary to 7.3.3. Do you see why?
Theorem 7.3.13 is much the same as Corollary 7.3.5, with "absolute convergence implies convergence" stuck on at the end.
7.3.8 and 9 parts b are obvious for the same reason as above; the terms don't go to zero.
Theorem 7.4.2 on alternating series is very important (and the proof could be expanded. Do you see why all those assertions are true?).
Remark 7.4.11c is really an important theorem (actually, several theorems), which is needed to explain 7.4.16, which is amazing. [This explains why we have to be very careful about the order of the sum of an infinite number of terms. This even bears on why we don't allow partitions for integrals to have an infinite number of points.]
Theorem 7.4.15, which says something about rearrangements, has a non-trivial proof.
Theorem 7.4.7 is Corollary 7.1.21, only stated employing a new term.

Jan. 14
Due Wednesday, Jan. 16:  Section 6.3: 2, 3b, 5, 6, (9 or 10, your choice)  [They are not assigned here, but 13, 14, and 15 are important named theorems studied in Math 505.]

Due Friday, Jan. 18: Section 6.4, page 262:  Don't do #1, but explain why is "a > 0" is necessary for #1. Do 3 (this is slightly different from the theorems in the text), 4, 6c, 10a

Due Wednesday, Jan. 23  Section 6.4,  page 263:  5, 7L, 10d,e,
                                        Theorem 6.4.6 is stated with c and d where many authors would use a and b (and vice versa). Restate it with the original integral from a to b and redo the entire proof to fit that statement. 

Jan. 4
Comments on Chapter 6:  Chapter 6 is on the definition of the integral. Definition 6.1.1 is important, as are Lemmas 6.1.3 and 6.1.5.  Note how often "sup" and "inf" appear on pages 243 and 244. When the sup of the lower sums and the inf of the upper sums agree, the function is Riemann integrable (Def. 6.1.6).  Theorem 6.2.1 is used to prove a function is integrable. The two major and simple cases are given in Theorems 6.2.2 and 6.2.4.  Be sure you understand those two proofs. (Finally, uniform continuity does some good!)  6.3 takes some time to prove properties of integrals. T6.3.1a takes more work than you might expect. T6.3.4 is important but not easy. Corollary 6.3.5b is interesting.
    We thoroughly cover 6.1, 6.2, 6.3 through 6.3.9, 6.4 (including the Fundamental Theorem of Calculus, two ways), and 6.5.

Due Friday, Jan 11:  Read 6.1. Do Section 6.1, page 245: 2, 4. Also, find a Calculus text and see how it motivates the definition of "integral."  On your homework name the text and its author and make a short comment on whether the motivation is solely finding the area under a curve or if there is some other motivation.

Due Monday, Jan. 14:  Read 6.2. Let f(x) = x on [0, 2]. Find a partition P such that U(f, P) - L(f, P) < ε.
    Also: Section 6.2, page 249:  7, 8 [pictured on page 130], 10

Remark 6.2.8 is far from trivial. T6.2.7 says there is one. R6.2.8 changes that to "all".  R6.2.8 can be useful if you already know that f is integrable. Then it tells you that any sequence of partitions with norm converging to zero will work.  We usually pick a sequence with n subintervals of equal width. By 6.2.7 alone we would not know that the equal-width-subinterval sequence would be the one that works.

Page 258, Theorem 6.4.4b (and Remark 6.4.5b) is extremely important. It is also called "The Fundamental Theorem of Calculus" (there is more than one version of the FTC). I will sketch an illuminating picture and give a proof. Be sure you can sketch the picture and give the proof. After all, this is the Fundamental Theorem of Calculus.

    In 6.3, Properties of the Riemann Integral, Theorem 6.3.4 is very difficult to prove and we will not do it. Furthermore, Corollary 6.3.5 depends upon that theorem and the proof of  fg being Riemann integrable (part b) is not simple, unlike all the proofs with fg that we have previously done. It really does use part (a) in a clever fashion.
    Then, Theorem 6.3.8 through 6.3.10 are of less use and we will skip them.

Typos in older printings that have been fixed in recent printings (ignore if your printing is number 5):  Page 247, in section 6.2, the last line of the proof of 6.2.1 in some older printings begins "Subtracting ....".  It should read, "Since L(P,f) and L(P,f)+epsilon are within epsilon of one another, so must be the upper and lower integrals, proving the desired result." 
    Page 257, section 6.4, Theorem 6.4.2:  If the third last line on your page begins "If  mk ...", the last three lines on the page are correct, but you could skip them (using Lemma 6.1.3) and resume on the next page.


 
Advice:
1)  Memorize all definitions we have used multiple times (in proper left-to-right order)
2)  Memorize the precise hypotheses and statements of all theorems we have used multiple times.
3)  Learn counterexamples to false statements that resemble results we have used multiple times, but fail to be true because some critical hypothesis is missing.
4)  Learn how proofs are written by
    a)  studying, in order to be able to reproduce, the types of proofs we have done multiple times.
    b)  studying, in order to be able to reproduce, the simpler proofs of major results (I do not expect you to be able to prove the most complicated theorems we studied)
    c)  Noting how they "follow the logic"
5)  Learn the basic logic we have used multiple times.

    For an extensive list of things you cannot do in proofs, see here.
    For an extensive list of things to do in proofs, see here.
 
Comments on proofs:  Proofs are sequences of prior results from which the statement to be proven is a logical consequence. The statement to be proven usually has some letters in it, and those are given and may be used in the proof. All others letters, such as bounds, ε, and δ, must be quantified or defined within the proof.  For example, usually ε is not mentioned in the theorem itself. You must "Let ε > 0" (for a representative case proof) before you can use ε in any other sentence. If you know {f(x)} is bounded, you cannot mention M for the first time by writing  "|f(x)| ≤ M, for all x."  You must, instead, say, "There exists M such that |f(x)| ≤ M for all x" and give a reason why. 

    When you read theorems, pay attention to the hypotheses. Note in the proof where and why they are used. Chances are a similar result is false if any hypothesis is missing. You should be able to think of counterexamples to similar statements with a hypothesis missing. Your attitude to mathematical statements must change. You should become skeptical.
    Some things you know well and are certain about. Good. Anytime something new and similar appears, you must develop the attitude that it might be false. Perhaps you can prove the new variant from your list of prior results. Then you know it is true. Or, perhaps it is false and you can prove it false with a counterexample.
    You need to learn how proofs work. Read the proofs in the book. See if you can anticipate where they begin and where they end. Every day your homework includes reading and studying the proofs of the theorems in the text. I will try to list which ones are more important and which ones less important so you can use your time efficiently. Nevertheless, it will be time-consuming. Previous students agree that Advanced Calculus required a lot of hours of work per week to do well.

    When you study a theorem, you must learn to recognize when it applies and when it does not apply.  If we change the hypotheses slightly, it is possible that the conclusion will no longer hold. Examples of similar "conjectures" are extremely valuable in helping you learn the precise meaning of a theorem. That is why, in Section 2.3, his problems are so good. Look at Section 2.3, page 86, problems 5, 6, 7, 10, 12, 13 which are all interesting, but not assigned at this time. Note 12 and 13 on page 80, which are similar and illuminate limits of products.


Homework notes:  If you are to prove an "iff" statement, there will likely be two proofs, one in each direction. When you do part, clearly label which direction you are doing. [I use "=>" and  "<=" to label the two directions].
    When you are resolving a conjecture, first clearly state if it is true or false. Then prove your assertion.
    On your homework, please put the section number and/or page number with each problem. Your homework often has problems from several different sections, and sometimes the same problem number in two different sections. Please identify each problem clearly. You will find this handy when you look over your old homework later. You can find the problems which you answered!



Syllabus:  Text: A Friendly Introduction to Analysis, Single and Multivariable, second edition, by Witold Kosmala. New copies are the 5th printing. If you bought a used copy of the text, it may be an older printing which needs many minor typographical changes. You can tell what printing you have on the copyright page just above the ISBN number where it says "10 9 8 7 6 5" or a similar string. The last number in the string is the printing number. Prof. Kosmala and I correspond and he has sent me a list of changes in the text, most of which have been corrected in the latest printing. Find them listed here (changes in second printing) and others here (changes in third printing, pdf format) and here (changes in first and second printings).
Class Hours: 11:00, MWF, Wilson Hall, 1-139.
Instructor: Warren Esty, Wilson Hall 2-238, 994-5354. westy AT math.montana.edu
Office Hours: 9-9:50 MWF and many other hours. You are welcome whenever I am in. I am more than happy to help. I love this material!

Prerequisite:  M-381, the first semester.

There is more about the class organization below.

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Homework. Every homework assignment includes reading the sections thoroughly, learning the concept images and concept definitions and understanding the proofs. I will note in class and here those parts of the text that can be skipped or skimmed.
One of each problem will be written on the chalkboard by some student (self-selected) before the beginning of class the day it is due. We will go over the homework in class at the beginning, using the work on the board for discussion, so homework must be turned in before class (unless you are using it to write your answer on the board, in which case you hand it in when you finish copying it). 
    If there are handouts, learn their content.
    Occasionally come a bit early and put a problem on the board.

The Course.  This is an "Advanced Calculus" course (which is the same as many junior-level courses titled "Real Analysis" elsewhere). In the first semester the emphasis of this course is only partly on mathematical results that are new to the students. The focus is primarily on learning the modern rigorous approach to mathematics in the context of calculus and deriving the results of calculus. The “calculus” component includes many familiar results. What makes the material “advanced” is the strong emphasis on proof and precision.

    The calculus component includes integration, infinite series, sequences and series of functions, and functions of two variables.
    The proof component includes proof techniques, concept definitions (as opposed to vague concept images), precisely stated theorems, conditional statements, hypotheses and conclusions, logic for mathematics, truth and falsehood, conjectures, counterexamples when statements are false, and rigorous proofs.

Grading.  Homework will count 200 points. Three unit exams will be 100 points each, and the final will be 200 points. Quizzes and class participation will count the remaining 100 points. Exam dates will be announced well in advance on this site.
    To receive full credit, homework must be handed in at the beginning of the class period when it is due. Homework handed in at the end of class or later will be regarded as late. Late homework will be accepted, but because the solutions will be discussed in class, late homework will receive less than full credit.
    You may cooperate with current students when doing your homework. Also, you make ask for help on individual problems from previous students. The homework I see from you will contribute to your grade and must be your work, or work done in cooperation with current students.
   
The final exam is 4:00-5:50 pm, Monday, April 29 (Monday of exam week).  Arrange your summer-break schedule so you can take the final at the scheduled time. You can find final exam times for other classes at: http://www.montana.edu/registrar/exams/

Etiquette. If you must miss an exam, you must inform me (Dr. Esty, 994-5354) well in advance. If you want a makeup exam I expect a very good reason, preferably well in advance.
   Cell phones must be turned off during class. Headphones or other electronic devices may not be used during class. As is obvious, students must behave so that others are not distracted during class.

Readings and Homework. The course will proceed straight through the text, with a few omissions. Homework assignments will be announced in class and listed on our website.