Math
362,
Advanced Calculus, Spring 2007
This website updated April 13,
2007. www.math.montana.edu/math361 [Note: URL with "361"]
Recent additions and changes
will usually be above the first
horizontal
line.
HW
due Wednesday, April 18:
Section 10.4, page 461: 4
HW due
Friday,
April 20:
Section 10.4, page 461: 5, 7, 8
(interesting!)
HW due Monday, April 23:
Section 10.5:
1, 3 (an excellent problem)
Friday, April 27:
Read
10.6. Read 11.1. Read 11.2. Do HW 11.2, page 489: 4, 5
The Final Exam is scheduled for May 10, Thursday of exam week, at
8:00-9:50.
The Final exam will be in two parts:
A 4-hour limited-time take-home exam given on the honor system will be
handed out at the end of class Friday May 4 and due at the beginning of
the
regular exam period, that is, at 8:00 am Thursday May 10 (Handing it in
early is
okay). A one-hour in-class exam will be
administered at 8:00 am, Thursday of exam week, May 10.
April 4:
Friday, April 6: University Day. No classes.
HW due
Monday, April 9: Skim
Chapter 9 and note the results mentioned below. Skim Section
10.1. Read Section 10.2 and do page 445: 1b,
3b.
Chapter 9 is "Vector
Calculus" which we will mostly skip 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.
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) 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. The
remaining results in 10.1 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.
Due Wednesday, April 11:
Section 10.2, page 446: 4.
no HW due Friday, April 13.
HW due
Monday,
April 16:
Section 10.3, page 454: 1a,b,c, 3a (use the definition, as he
requests. Omit 3b).
Section 10.4, page 460:
1a,b,c
Exam, Wednesday,
April 4, on
Chapters 7 and 8.
No HW due Monday, April 2.
Due Wednesday, March 28: (3.5)
8, page 367.
Friday,
March 30: Section 8.6, page
373:
1a,b [(a) and (b) are worth memorizing],c,g, 2a,b [For some
of these there is a distinctly easier way
than taking the derivatives.]
Due
Wednesday after Spring Break
(March 21): 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, although our examples for
series are mostly increasing. For 5d, keep all the other hypotheses but
have the fucntion on (0, 1) instead of [0, 1].) [Look at 8. This
is
important and
discussed at length in measure theory, in Math 547, Real
Analysis.]
[Dates after March 21 are likely, but tentative]
Due Friday, March
23: Section
8.4, page 357: 1a, b, e,
h, i, j, 5, 11 (long and important), 12a-f
(important. Do not do part (g)) [Also look at 8 and
15]
Due Monday, March 26:
Section 8.5, page 366: 1, 2, 3a,b,c, 5
[typo, it is supposed to
say "≤ ", not "<"],
6 [HW 8 is good, but not assigned.]
Mar. 7
Due Friday, March 9:
Section 8.2, page 346: 3, 4, 5, 8a
March 10-18:
Spring Break. No classes.
March 2.
HW due Monday, March 5 (revised):
Do the problem on a more-general limit-comparison text that was handed
out in class. That is, see if the hypotheses of positive terms in
Theorem 7.2.7 can be removed. Fix it, if possible.
HW due
Friday, March 2:
"Fix" the conjecture about Sn2/n2
->
c => Sn/n -> c by adding in a nice hypthesis on
the Xis (your choice) and then proving the revised
conjecture is true.
HW formerly due Friday, March
2, is now due Wednesday, March 7:
[:
Do these parts of #2 using examples on [0,
1]: 2b, c, d, e, g, h, j
Feb. 26
Chapter
8 comments: All results for sequences will be converted
to results for series. The sequence results will refer to the sequence
of partial sums.
Definition 8.2.1 of uniform convergence is 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 continous 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 hyptothesis. 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 we will not use (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 Wednesday, Feb. 28:
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]
HW due Wednesday, March 7 (postponed):
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.]
Feb. 22.
Due Monday, Feb. 26:
Section 7.4, page 322: 2b, c, 14 (be clear)
In class: Conjectures about Sn -> c and Sn2
-> c.
Feb. 14.
Comment: Theorem 7.3.13 is
much the same as Corollary 7.3.5, with
"absolute convergence implies convergence" stuck on at the end.
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.
Theorm 7.4.7 is
Corollary 7.1.21, only stated with a new term.
Due
Friday, Feb 16: Section 7.2, page 309: [show the test] 1a,
c,
h,
j, n, 2, Section 7.1, #16, using tools from Section 7.2.
Monday is a holiday, President's Day. No Classes.
Due Wednesday, Feb. 21:
Section 7.3,
page 315: 2, 7a,i, 9a,b,c
Due Friday, Feb. 23:
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).
Wednesday, Feb. 21:
Lecture, not from the
book, on special functions defined using integration (log, sine,
cosine).
Later: Section 7.4, page 322: 2b, c, 14
Friday, Feb. 9:
Lecture on Chapter 7. No HW due.
Monday, Feb. 12: Exam
on Chapter 6. (Day
changed to Monday)
Due Wednesday, Feb. 14:
Chapter
7.1, page 300. 1a,c, 3, 4, 5, 10
Due Friday, Feb. 2:
Section 6.3, page 254: 3b, 5, 7,
Due Monday, Feb. 5:
Section 6.3: 13 [for 13: Take the
hint in the back. It is hard.] [look at 6, 9, 10, 14, 15 -- 14
and 15 are famous results]
Section 6.4, page 261-263: 1, 3, 6a,b,
10a,d,e
In Chapter 6, Integration, we skip the
optional sections 6.6 and 6.8. Section 6.7 is review.
Wednesday, Feb. 7:
Section 6.4, page 263: 7L, 10f.
Section 6.5: page 270: 2, 3,
9a
Section 6.7 (Rules: Say if it is true, and if it is
false, give a counterexample) page 283: 2, 3, 4, 8
We begin Math 362 with Chapter 6 of Kosmala: Integration.
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 Wednesday, Jan. 24:
Section
6.1, page 245:
2
Due Friday, Jan. 26:
Section 6.2, page 249: 7, 8 [pictured on page
130], 10
Due Monday, Jan. 29 :
Section
6.3, page 254: 2
Comments on Chapter 6,
Integration.
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 Theorm 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
Theorm of Calculus.
Page 247, in section 6.2, the last line of the proof
of 6.2.1 in some 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."
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.
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.
Syllabus:
Text: A Friendly Introduction
to Analysis, Single and Multivariable,
second edition, by Witold Kosmala.
If you bought a used copy of the text,
it may be an older printing and need many minor changes. 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). Please correct your text
now. Some changes refer to the first printing
(and, if you have the second, those may already be corrected). Other
changes refer to the second and third printings. The text is in the
third
printing. You can tell which printing you have on the "Library of
Congress" page right after the title page. Look above the ISBN number
for the countdown 10 9 8 7 6 5 4 3 2. If it ends at 2, you have
the second printing. If it ends at 1 (like mine) you have the first
printing.
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: 9-10 MF, 9-11 TuTh, and many other hours. You are
welcome whenever I am in. I am more than happy to help. I love this
material!
************
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.
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.
The Course. This is an "Advanced Calculus" course. The
focus is primarily on learning the modern rigorous approach to
mathematics
in the context of calculus. The “calculus” component includes many
familiar
results. What makes the material “advanced” is the strong emphasis on
proof
and precision.
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. .
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.
Calculators. Calculators play no role whatsoever in
this
class. This is not a computation class.
Etiquette. If you must miss an exam, you must inform me (Dr.
Esty, 994-5354) well
in advance. I rarely give makeup exams and I do so only for very good
reasons
approved 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 some omissions. Homework assignments will be
announced in class and listed on this website.
Topics:
Second semester resumes where Math 361
left off.
Chapter 6, Integration.
Chapter 7, Infinite Series
Chapter 8, Sequneces and Series of Functions
parts of Chapter 9, Vector Calculus, Chapter 10, Functions of two
variables and Differentiation, and Chapter 11, Multiple Integration.