Math 382, Advanced Calculus, Spring 2012
This website updated April 24, 2012 (the URL uses M381)
Recent additions and changes will usually be above the first horizontal line. Course policies are here.

No HW is due Wednesday April 25 or Friday, April 27: We will review in class. Look at last year's final exam (handed out in class Monday) and come prepared to ask questions.

The final exam is 8:00-9:50 am, Friday, May 4. 
 (You can find final exam times for other classes at: http://www.montana.edu/registrar/pdfs/FinalsSchedule.pdf )

April 19
HW due Monday, April 23: Section 11.1: 8

April 18
HW due Friday, April 20:  We talked about ratings based on a Bradley-Terry model when P(A beats B) is a/(a+b) where a and b are associated numbers, but not their "ratings". Their ratings were "k ln a" and "k ln b", where k was adjusted to make the probability of 3/4 correspond to a 200 point difference. k = 182 worked. We decided that *if* we wanted changes to be of the form "k1(W - L) + k2(the sum of the difference in ratings over the games)", [i.e. up k1 for a win plus some multiple of the difference in ratings, which could be negative] then we could find the ratio between the constants and adjust their sizes to fit how fast we want ratings to change. But, is that form (in quotation marks above) for the changes justified? We used maximum likelihood estimation to find the a/b ratio (which determines the probability). Derivatives are useful for discussing change. Play around with the equations we already have and see if you can find a way to change ratings or change a and b that is justified. Don't search the web. Hand in something, but don't spend long on it. If this isn't fun, don't spend long on it.  

April 13
HW due Wednesday, April 18:  Section 10.6: 12
April 13
HW due Monday, April 16:  [Based on material covered in class Friday.]  Use the gradient and "the method of steepest ascent" to maximize the log likelihood we derived in the case when the the data is as follows. Begin anywhere, or if you don't want to pick your own initial point, with, say, a = .5 and b = .45. Change (a, b) using a small multiple of the gradient and then redo it. Repeat until the answer stabilizes.
 [Do this with a calculator or computer, but do not use a powerful equation-solving program or function-maximizing program.]

A
 B
 C
 D
1
 A*
 B
 C
2
 A
 B
D*
3
B*
 C
 D
4
 A*
 B

5
B*
D
6
 A
 B*
D
The six voters are listed in the left column. They vote for the "best" movie. The four movies they vote among are in columns. The ones they actually saw are listed in the row, and their favorite among the ones they saw is starred.
    The model:  Assume the probability is px that a voter who has seen all four movies votes for movie X. If the voter has seen a subset of the movies, the probability is px divided by the sum of the probabilities of the movies the voter has seen, if X was seen, and 0 otherwise.
    Because no one voted for C, we eliminate C from the computations (as if C's rating were 0). Then, because the probabilities sum to 1, the probability for D is a function of the probabilities for A and B.
    The likelihood function to maximize in this example is a2b3(1-a-b)/((a+b)2(1-a)2). We decided to maximize its log instead, using the method of steepest ascent, in the region where a and b are probabilities such that a+b < 1.

April 11
HW due Friday, April 13: More on ratings. [Please do not use the internet, but working with friends is encouraged. Do not spend more than two hours unless you just want to.] I would rather you followed up on any ideas you had before. Maybe you will discover a new method with some virtue! Oddball ideas with no good resolution will be worth full credit (or more!)! Just show me some ideas!
    However, if you want, instead, you may follow up on the idea that P(A beats B) is estimated by RA/(RA + RB), where R is the rating. Try to figure out some (simple) way to update the ratings based on the results of future games. One way to change A's ratings would be to add  a constant number for each win (subtract for each loss) and then add (subtract) a constant times the difference in ratings for each game played.  The constants need to be well-chosen. Can you figure out how to choose them to maintain stability? (Say, in the probability = 3/4 case?) Or, maybe you can figure out some other justified way to change ratings.
    Now, extend this to three (or more) teams/players. How would teams be rated based on win-loss records between them?
    We are just about ready for an example with numbers. If you are ready to give one, do so.

April 9
HW due Wednesday, April 11: More on ratings and chances of winning.
        The goal is to assign numbers ("ratings") to 3 or more teams/players, where the numbers correspond to the probability of winning in a two-player contest.
        If you off to a good start in any context, continue with it, in which case you may ignore the following.
        If you don't like your start you may consider three players who have played two-player contests, each contest ending simply in a win or loss.  e.g. A has played B 4 times and won 3. B has played C 3 times and won 2. What does the system predict for A playing C?  What ratings do the three players have in your system?  If then A plays C and A wins, how do the ratings change? If C wins?
        Goal:  If three players have "stable" ratings which predict that A will beat B n times out of m and they play m games and A really does win m, we want the ratings to not change.

HW:  In addition to creating some theory, show
some example numbers (ratings)for 3 or more players, and
example future games with their wins and losses and
how the ratings change due to those games

        Your system does NOT have to be a good one yet. There is no "right" answer. I would love it if you would follow up on some oddball system -- maybe it will lead somewhere!
        --  Warren Esty
April 2
HW due Monday, April 9:  [No internet searches allowed. Working with classmates is encouraged. Don't spend more than two hours unless you just want to. This is a multi-day project.] Hand in something--anything--along these lines.
    For some game or sport, when two players or teams play twice, the outcomes are not always the same. We would like to rate them in some way that somehow allows us to approximate the likelihood of one team beating (or drawing) another in an upcoming contest. Think of some--any--rating system. [Your choice of game or sport.] We will try to develop one or more rating systems that have nice properties:  previous games played determine the ratings, future games may change (adjust) the ratings, and the ratings of two teams are related to the chances that one team would beat (or draw) the other if they played.

April 2
HW due Wednesday, April 4:  none

HW due Monday, April 9:  To be announced Wednesday, April 6.


March 27
The final exam is 8:00-9:50 am, Friday, May 4.  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/pdfs/FinalsSchedule.pdf )

Theorem:  Let f: R2 -> R1. If f is differentiable at (x0, y0), then fx(x0, y0)and fy(x0, y0) exist.
This can be used to show a function is not differentiable at a particular point (x0, y0), but will not suffice if fx(x0, y0) and fy(x0, y0) exist.
Theorem:  If fx(x0, y0) and fy(x0, y0) exist, define ε [a function of (x, y)] by
*    f(x, y) = f(x0, y0) + fx(x0, y0)(x - x0) + fy(x0, y0)(y - y0) + ε ||(x-x0, y-y0)||.
Then, f is differentiable at (x0, y0) if ε -> 0 and
f is not differentiable at (x0, y0) if ε does not go to 0.
   Regardless of whether f is differentiable or even continuous, there is an ε such that equation * holds. If (x, y) is not (x0, y0), equation * can be solved for ε. The theorem above is very similar to the definition of "differentiable"--it just notes the role of the partial derivatives.

HW due Friday, March 30:  Section 10.4, page 460: 1a,b,c, 2, 4

HW due Monday, April 2: Section 10.5, page 464: 1, 3 (an excellent problem), 6, and use level curves to sketch a graph (picture) illustrating a polynomial function R2 -> R1 which has a strict local minimum at the origin along every line through the origin but the origin is not a local minimum.

March 19
    Chapter 9 has the dot and cross products (9.3, especially T9.3.3) which are used in differentiation, parametric equations (9.4, especially of lines, 9.5), and planes (9.5) which are used in differentiation (especially 9.5.3), and arc length (9.7).

Many of the results for functions of two variables are close parallels to the corresponding results for functions of one variable, but some are not close parallels. We will pass rapidly over the ones that are close parallels, and dwell on the ones that are not.
    We do not need much of 10.1. Anyway, the results through 10.1.8 are close parallels to what we have studied. 10.1.10-10.1.13 we don't need.
    In 10.2, we use 10.2.1 and the notation above it, 10.2.6, 10.2.7, 10.2.8 10.2.9, and 10.2.13, which are all close parallels to things we know.  10.2.10 is a topological way of dealing with continuity, and 10.2.14 is also a topological thought, both of which we will not use.
    Examples 10.2.2 and 10.2.4 are complicated, but illustrate the types of functions that appear in this subject. In place of 10.2.4, I will do  xy/(x2+y2 ) which is a simpler, and therefore better, example.  He finds another reason to use this simpler function in 10.3.2c, which is a major example.
   10.3, "Partial Derivatives," is where the action begins.  Some things are complicated and long, but they are important. Be clear that 10.3 is not yet about differentiation (10.4 is "Differentiation."). 10.3 is about partial differentiation, which is quite a bit simpler than differentiation. I will clarify 10.4.1 through 10.4.6 when we get there. 
    We will wait to do 10.3.4 until after 10.4.5. Do 10.3.5 instead. 10.3.4 has too much overlap.

HW due Friday, March 23:  Read Section 10.2 and do page 445:  1b, 3b

HW due Monday, March 26:  10.2, page 446:  4, 8 (a reason, but not a formal proof required), 14

HW due Wednesday, March 28:  Section 10.3, page 454:  1a,b,c [for part c), change the value to 0 {from 1} at (x, y) = (0, 0)], 3a (use the definition, as he requests. Omit 3b).

Due later:
Section 10.4, page 460:  1a,b,c
Section 10.5:  1, 3 (an excellent problem)
Read 10.6.  Read 11.1.  Read 11.2.  Do HW 11.2, page 489: 4, 5

March 19
HW due Wednesday, March 21: I will ask you in class Wednesday, as a long quiz, to prove that if a power series about 0 converges at x0 and |x| < |x0|, then the derivative of the power series at x is given by the power series of term-by-term derivatives (much like 8.5.15b, but without the use of R), given only results from 8.4 and earlier. That is, learn all the results after 8.4 up to 8.5.15b that are necessary for its proof and string them, with proofs, end-to-end. Omit the results that are not directly in the line of results from 8.4 to 8.5.15b.
    So, your homework, which is not to be handed in, is to first list in outline form the results (theorems) necessary to get from 8.4 to 8.5.15b (omitting the ones that are stated in the book but not necessary for this limited task) and then prove them, in order.
    Then, without your notes, you will be asked to repeat this as a quiz.

March 2
HW due Wednesday, March 7:  9.3: 16, 17   [Comments below on 9.1-9.7 have been expanded. Look at them again.]
Also, List some of what you learned in differential calculus (Now M 171-2) that is not "how to do calculations." (Work with a friend to expand your list.) That is, if I gave you a calculation device that could do all the derivatives, takes all the limits, etc., what did you learn that that device cannot replace?  [Note:  This list will help us identify what there is to learn about multivariate calculus.]
    If you have a graphing calculator, bring it Wednesday. We will play with some interesting parametric equations.

HW due Friday, March 9:  9.5:  1, 2, 5, 6

March 10-18 is Spring Break.

HW due Monday, March 19:  Skim Section 10.1 (We don't study "connected").  Do 10.1, page 436, 4, 7, 8
Feb. 28
Last year I did not give an exam devoted solely to Chapter 8. Here are the problems that come from Chapter 8 on last year's exams.

Monday, March 5:  Exam on Chapter 8.

HW due Wednesday, March 7:  List some of what you learned in differential calculus (Now M 171) that is not "how to do calculations." (Work with a friend to expand your list.) That is, if I gave you a calculation device that could do all the derivatives, takes all the limits, etc., what did you learn that that device cannot replace?  [Note:  This list will help us identify what there is to learn about multivariate calculus.]

Chapter 9 is "Vector Calculus" which we will cover rapidly so we can move on to Chapter 10, "Functions of Two Variables,"  where we discuss differentiation in multiple dimensions. Nevertheless, Chapter 9 has some interesting and important results.
 
9.1.1 The Pythagorean Theorem (distance) in 3-D
9.2.4  perpendicular, parallel, the "direction" of a vector
9.3.1-3  dot product, properties, angle between vectors
9.3.8-12  cross product (notation typo in old printings: 9.3.9b should, obviously, have "x" [cross] for "+")
Last line above "Exercises 9.3" has a formula for area (I don't know why this was not labeled as the theorem).
9.4 and 9.5  parametric equations (9.4, especially of lines, 9.5) which are used in differentiation (especially 9.5.3),
Example 9.4.2 generalizes to 3-D.
9.5, amazingly, has an asterisk on it ("9.5*")which suggests it is optional. Not so. Theorem 9.5.3 is critical, since it gives the equation of a plane in 3-D, and the definition of "differentiable" refers to approximation by planes.
9.6  mostly very parallel to previous material. It treats a function in 1-D with 3 images (such as a parametric representations of a 3-D path over time) as 3 functions (one for each dimension), each with one image. Theorem 9.6.13 "The Fundamental Theorem of Calculus" is a perfect parallel to the 1-D version we already studied.
9.7  arc length, Suppose a particle is moving in 2 or 3-D. We describe the path parametrically (with time as the parameter). The distance it moves is arc length.

Many of the results for functions of two variables are close parallels to the corresponding results for functions of one variable, but some are not close parallels. We will pass rapidly over the ones that are close parallels, and dwell on the ones that are not.
    We do not need much of 10.1. Anyway, the results through 10.1.8 are close parallels to what we have studied. 10.1.10-10.1.13 we don't need.
    In 10.2, we use 10.2.1, 10.2.6, 10.2.7, 10.2.8 10.2.9, and 10.2.13, which are all close parallels to things we know.  10.2.10 is a topological way of dealing with continuity, and 10.2.14 is also a topological thought, both of which we will not use.
    Examples 10.2.2 and 10.2.4 are complicated, but illustrate the types of functions that appear in this subject.  In place of 10.2.4, I will do  xy/(x2+y2 ) which is a simpler, and therefore better, example. He finds another reason to use this function in 10.3.2c, which is a major example.
   10.3, "Partial Derivatives," is where the action begins. Some things are complicated and long, but they are important. Be clear that 10.3 is not yet about differentiation (10.4 is "Differentiation."). 10.3 is about partial differentiation, which is quite a bit simpler than differentiation. I will clarify 10.4.1 through 10.4.6 when we get there. 



Feb.21
Comments on 8.5 and 8.6.
T8.5.5 is awkwardly stated. The key is the distance between the x-value for which it converges and the "a" in "x - a". That is,
If it converges for x0 and |x - x0| = d, then it converges for all x such that |x - x0| < d.
C8.5.6 follows by contrapositive.
Definition 8.5.7 is similar to a theorem which says "There exists R such that ....".
T8.5.8b  should say (and does in recent printings) "≤ r", not "< r".
T8.5.9 is the old Limit Ratio Test redone in terms of the coefficients (ak used be a term; now it is a coefficient on a term).
T8.5.11 follows from T8.3.1.
T8.5.12 may be omitted now (but keep it in mind if you ever need to work with an endpoint).
T8.5.13 is a useful Corollary.
T8.5.15 is major. It is largely why we have put so much effort into series.
T8.5.15a:  the integral of the sum is the sum of the integrals.  [This is really about definite integrals.]
T8.5.15b:  the derivative of the sum is the sum of the derivatives.
T8.5.16 is used to prove 8.5.15 and belongs first.
Remark 8.5.18 follows T8.5.16 and is relevant to T8.5.15.
C8.5.19.  Call both functions f (not f and g).
T8.5.20 I have actually used this, but it is too subtle for us here.
T8.6.1. My printing has a significant error at the end of page 370. It recalls Taylor's Theorem (5.4.8) and says the "remainder" is the tail of the series. No. This works for T8.6.1, but not in general. The "remainder" was the difference between the function being expanded and the polynomial approximation of it. If (and only if) that difference converges to zero the remainder equivalent to the "tail". We had one example, (page 211, Example 5.4.3, repeated as Example 8.6.2, page 371) where the power series did not converge to the function and the "remainder" of T5.4.8 is not the "tail" from page 370.
T8.6.3.  Restated: When is the Taylor Series of a function equal to the function itself? When the remainder has limit zero.


Feb. 16
HW due Friday, Feb. 17: 8.4: 8, 12d,e,f (important. Do not do part (g)), 15a-d.
    Be sure you understand how the series theorems are really sequence theorems, rephrased to fit the new context.  

Monday, Feb. 20 is a holiday.
 
HW due Wednesday, Feb. 22:  (8.5) 1, 2 [Do part (a) last; it is the hardest.]

Friday, Feb. 24 I will be out of town at a meeting.

Think of T8.5.16 before T8.5.15, not after.

HW due Monday, Feb. 27:  (8.5) 3a,b,c, 5 [typo in some printings; it is supposed to say "≤  r", not "< r"], 6a (only part (a))   
.
HW due Wednesday, Feb. 29 (Leap year!):  Section 8.5, page 366:  8b, 11, 14b,c

HW due Friday, March 2:  Section 8.5: 13, 19

Feb. 7
HW due Monday, Feb. 13:  Section 8.4, page 357:  1a, b, e, h, i, j, 2b, 5,
    We primarily use the Weierstrass M Test (T8.4.12). Do not worry about T8.4.13-14 now; they are too subtle. T8.4.15 is major. Compare it to Theorem 8.3.3.

HW due Wednesday, Feb. 15:  Section 8.4: 11 (important), 12a,b,c

HW due Friday, Feb. 17: 8.4: 8, 12d,e,f (important. Do not do part (g)), 15a-d.   


Friday, Feb. 10:  No HW due. Exam on Chapter 7 (not material in Chapter 8).  Here is an old exam that covered Chapter 7 but also covered part of Chapter 8 and even things we did in class related to Chapter 6. You are still responsible for the Fundamental Theorem of Calculus and the rest of Chapter 6.

Feb. 2
HW due Monday, Feb. 6: 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]
                                                    Section 8.2, page 346: 3, 4, 5, 8a

HW due Wednesday, Feb 8: Section 8.3, page 350:  2 (Do not use 8.3.4), 3, 5 (for part c, he uses increasing but you may consider the decreasing case instead if you prefer. I find it more natural to consider decreasing to the zero function.) Also 8. [This is important and discussed at length in measure theory, in Math 547, Real Analysis = Measure Theory.]

Jan. 27
Due Friday, Feb. 3: 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 [not Section 7.2, page 310, 5, 7, as previously listed]

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 of a series and therefore apply to series.
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).

HW due later [date to be determined]:  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]
                                                    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, he uses increasing but 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.] 

Jan. 23
HW for Wednesday, Jan 25:  Feel free to stop after two hours. Please make some conjectures (easy ones) and resolve them. The handout you got in class is here.

Due Friday, Jan. 27:  Section 7.1, page 302, 20. Section 7.2, page 309: [show the test] 1a, c, h, j, n, 2

HW due Monday, Jan. 30:  Section 7.2, page 310, 5, 7, Section 7.3, page 315: 2, 7a, 9a,b,c 

Due Wednesday, Feb. 1:  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.

Jan 10
Math 382 is a continuation of Math 381. We resume where we left off.
This page has retained from last semester's page all the general advice about doing proofs and some comments on Chapter 6.

Due Friday, Jan. 13, 2011:  Correct your final exam, which was returned Wednesday, hand it in with the original, and come with questions about it. Rewrite Theorem 6.4.6 (Integration by Substitution) so the left side is integrated from a to b and the right side is stated correctly with a better choice of letters. Then prove it cleanly using your new letters. (make it really nice!)

Monday, Jan. 16 is a holiday. No classes.

Due Wednesday, Jan. 18:  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:  7L, 10d, f, 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.

Due Friday, Jan. 20: Section 6.5:  page 270:  2, 3, 9a, b
                             Section 6.7 (Rules:  Say if it is true [no proof required], and if it is false, give a counterexample)  page 283: 2, 3, 4, 7, 8

Due Monday, Jan. 23:  Section 7.1, page 300: 1a, c (this is telescoping; find the sum), 3, 4, 8, 10

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, 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.  
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 with a new term.



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 (eventually, some of it next semester) 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.

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: 9-11 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 fucntions 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 8:00-9:50 am, Friday, May 4.  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/pdfs/FinalsSchedule.pdf )

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.