Math4606SS05.

Math 4606, Summer '05

Last changed: August 6, 2005.

This course is a proof-oriented version of one- and several-variable Calculus.

The course meets 7 hours a week:
11:15-12:05 MWF, and
11:15-1:05 TuTh (break: 12:05-12:20), always in FordH B10.

Our TEXT is Advanced Calculus 3rd ed, by Taylor & Mann
ISBN 0-471-02566-6; John Wiley & Sons
DON'T buy Buck's Advanced Calculus!

The text may be late arriving, but that doesn't matter much; we will
start wih some material on Logic and Sets that is not in the text anyway.

Office Hours: 12:20 - 1:05 MWF, as Help Sessions in FordH 10.
also, "on demand": just after class TuTh. Otherwise, by appointment --
or call 5-3855 to see if I'm in and available, especially before class.

You can ask questions by email too: jodeit@math.umn.edu
Please, send me messages in TEXT mode only, if you can! Also, here is a Web link to
a list of special codes for math symbols (most of them start with the \ character).

Announcements, hints, "news" will be here.

Links to PDF documents

News August 6: You grades should be available to you
by 9:38pm tomorrow, August 7. If you cannot access your grade, please send me email!
I wish all of you well. And should you have trouble dealing with proofs
in later classes, send me email or come to an Office Hour. I'll miss you.

News August 5: The list is gone. The next News will
have the time I submitted your grades to the Registrar, and
when the Registrar's site said you could access your grade online.

News August 4: Here is the list.
If you checked it before 11:35pm, check it again!
The HW gradelines follow it.
The list will be removed tomorrow afternoon.

CODE, GPA, SP7, HW6

HW Gradelines:
p-top 198
A 158
B 126
C 101
D 81

News August 3a: "Some Solutions" has the start of a solution
of SP 7; more later, as well as info on the Final.
Info on the Final Exam, parts A and B
* Each Part will have 10 questions, But only 8 will be scored!!
* Questions will be based on HW, Quizzes, SPs, with changes made
but with the same methods used. Up to 2 questions can be "new."
* In effect, each part will essentially be 2 Quizzes. However, one of
the questions in each part will be worth 20 points -- all the others being 15.
* Please line up as usual, and the Final can begin sooner. If you ALL
agree, we can start tomorrow after a shorter break.

* Proofs will be very important on the Final!

Advice: get plenty of rest, bring water, loosen up by asking questions
in the first half and trying to answer those of others...
Good Luck!

Questions about the Final Exam(s)

"I was just wondering if it is possible to have to duplicate proofs of
theorems done in class on the final."
** I think so -- at least you can expect to have to "do" proofs
that were done in class. But you could do a proof from scratch; it does
not have to be a duplicate. Sometimes you can use a Theorem without
proving it, unless you're asked for a proof of it. You have to identify
Theorems you use, and show that you have checked their hypotheses...

"I thought you said that we would only be tested on the Implict Function
Theorem and the Inverse Function Theorem on either the final or on the last
quiz, but it would only be one on each. Is that right?"
** No. That was just for Quiz 7.

News July 31: Assignments have been updated:
"Assignment 7" and Special Problem 7.

News July 26a: "Some Solutions" has more
solutions: Quiz 6 #3, Sec 6.4 #3(b), Sec 6.52 #7;
the latter are (1) and (4) from HW 5.

Questions about Quiz 7
What sections will be covered on Quiz 7?
Sections 12.1, 12.2, 12.4, 12.5, 12.6 (statement and norm facts used in
the proof of Assertion 1 of Theorem VIII), "differentials" (6.4-6, 12.1-7),
19.0, 19.1, "telescoping series," 19.2, 19.3 include the Quiz material.

News July 25: Assignments have been updated:
Assignment 6 and Special Problem 6.

News July 20: Questions about Quiz 6
What sections will be covered on Quiz 6?
Sections 4.5, 5.2, 5.3, 6.0, 6.4 (n-dimensional version; i.e., f maps
n vbles to 1 vble), 6.5 (done today), 12.1, 12.2, 12.4 include the
Quiz material.

News July 18: Assignments have been updated:
Assignment 5 and Special Problem 5.

Today we proved the Chain Rule for functions from \r^n to \r^m to \r^k.
Along the way, we used Special Problem 5.

News July 15: Be sure to send me your homework questions today
or this evening. Weekend plans will keep me away from my computer until
Sunday evening, when I'll look again.

Yesterday we began work on Chapter 12. We have to skip over the more leisurely
presentation in Chapter 6. Today we spent almost all the time on yesterday's Quiz.

The exception was the Difference of Powers Formula,
b^n - a^n = (b-a)\sum_{k=0}^{n-1} b^{n-1-k}a^k, proved by
writing the right-hand-side as the difference of two sums, the first with an
extra power of b, minus the second sum with an extra power of a.
Then we canceled some terms, easily done after an "Amelia Bedielia" change of
summation index in the second sum.

In Chapter 12 the DEFINITION of "differentiable at a point in R^n" is on page
336.8, starting with "But the" at the end of a line, ending at (12.1--2) on the next page.
Please look at the short paragraph at 337.33.
At various places you'll find "script-L(\r^n, \r^m)" and that just means "m x n matrices"
as far as we are concerned. You might want to look at Sec. 11.2 -- the notation is defined
there, tho it relies on Section 11.1 (I suspect you know about the 11.1 stuff).

Our next major piece of work is the Chain Rule, Theorem II, page 338.96.
It is important to note, just after (12.1--10), that the function G(k), defined
for k NOT zero, will later be defined to be zero when k IS zero
(page 340, line 2). This makes G(k) continuous at k=0.

News July 13: Quiz 5 is tomorrow. Today we concentrated on Sec. 6.4,
defined "differentiable at a point in R^n" and proved that "diffable implies
continuous (at that point)" and that "diffable" implies that the coefficients in the
definition of "differentiable at a point in R^n" are the partial derivatives
of the function at the point of differentiability. We skimmed Sections 6.1 - 6.3, noting
their intuitive and "explanatory" content: well worth looking at.

HW2 was handed back today, and HW3 will be given back tomorrow.

News July 12: A very wordy solution of Special Problem 3
has been added to the Special Problem solutions document.

Sections 16.31 and 16.41 are about sequences in one and in higher-
dimensional spaces; 16.31 also introduces the concept of subsequence.
The higher-dimensional version of the Nested Intervals Theorem is mentioned
in Section 16.4. We have already covered a simplified version of the Heine-
Borel Theorem, Theorem VII in Section 16.6. Our version is not a
"watered-down" version! Theorem VII in Section 16.6 is an almost immediate
application of our version.

News July 11: Assignments have been updated:
Assignment 4 and Special Problem 4.

News July 9: Homework correction! In (1) assume, in
addition to the other conditions, that f(x) is continuous at x=0.

News July 6: By request, a solution of Assgt 2, (2) has
been posted. Other interesting requested solutions will be added
to the new document "Some Solutions."

News July 5: I hope I left my textbook in my office! So I had to
invent some questions for next Monday's homework. The new Assignment is now posted.

News July 3: "Sequence Theorems" has some Theorems
about sequences. Those all involve limits so far. The Theorems also
serve as examples of proofs.

News July 1: "Details" has details on parts of Rolle's Theorem that
were treated only lightly in class.

News June 29: Quiz 3 tomorrow, Thursday June 30.

The proof of the boundedness theorem, using the Bolzano-Weierstrass Theorem, that
was given in class, is posted. Next time, we complete the proof of the Intermediate Value
Theorem, and perhaps you will have some questions. If not, we will review derivatives
and the Mean Value Theorem so that we can begin Chapter 4.

News June 28a: (Asked about after class) Taylor and Mann sometimes use the term
finite closed interval. They mean bounded closed interval.
Their term is very rarely used nowadays, so it looks strange, because the only
closed intervals that are, literally, finite consist of one point or are empty.
The term was in common use a long time ago, though.

A solution of Special Problem 2 is posted.

News June 27: Assignments have been updated.
We have started working in Chapter 3. The Limit Theorems we are working
on are in Section 1.64. You may be asked to prove limit theorems for
sequemces as well! There, the large K corresponds to the (small)
positive delta...

News June 22: Quiz 2 is tomorrow, second half second period.
Please include study of the last Quiz in your preparation. I am liable to put
a problem that you had trouble with on a Quiz on a later Quiz!
Stuff about sequences is in section 1.62. Only the sequence stuff done in
class will be on tomorrow's Quiz.
Tomorrow I hope we finish our Chapter 16 detour and at least cover the intro section of Chapter 3.

News June 21: We are taking a detour thru part of Chapter
16. The material involving boundary points, however, is not
in the part of Chapter 16 we'll look at, namely 16.2 and 16.3. We may
have to discuss 16.1 too, so you may as well read that. It's important
that we work on the Theorems in Chapter 3. Even tho we'll prove some
more Theorems from Chapter 2, especially 2.5 and Theorem IV, Chapter 3
is where the Advanced Calculus starts. We've just been developing tools
so far.

News June 20: Assignments have been updated.

News June 15a: Axioms for the real number system: available below.
We have a papergrader. The first HW assignment will be due Monday June 27. It
will be a little longer than a "normal" assignment because you'll have more time
to work on it. Reminder: Quiz 1 is Friday, and Special problem 1 is due Thursday.
The first HW assignment is now on the Web.

News June 14: An Assignment sheet has been posted. It
has Special Problem 1, and "special directions" for Special Problems.

News May 20: A syllabus has been posted. The link above will take
you to the PDF links below.

News May 19: I'll be away until June 7, and will try to find places in
Paris that let me get on the Web, but I won't make it a major mission.

The course will have a Quiz each week, the first one the first Friday, the rest on
Thursdays during the second class period. Office Hours on Wednesday, Friday
would be good times for you to ask questions about homework, which will be due
Mondays. A major thing I want is to get you well-started on reading the text on your
own! In addition to homework (scored by a Papergrader) there will be Special Problems.
Special Problems are essentially Term Papers, and are scored by me.

The Syllabus will be posted by tomorrow evening.

Links, if any, to other Web pages pages related to this course will go here.
They are PDF files (you need Adobe Acrobat Reader to view them).

Follow this link to download the latest Acrobat Reader from Adobe at no charge.
"Express" URL

Math 4606 SS '05: Syllabus. 6/14

Math 4606 SS '05: Assignments. 7/31

selected Special Problem solutions 8/3

Some Solutions 7/26

Sequence Theorems 7/3

DETAILS 7/1

Class notes on the Boundedness Theorem 6/29

Axioms for the Real Numbers. 6/15

Below: background material from a much earlier version of this course

Introduction

Mathematical Logic

Sets, variables and quantifiers