Fall 2017 
General Information for Math 4031Section 1. 
Time 
9:30  10:20 AM, M W F. Our class meets from Monday August 21, 2017, through Friday, December 1, 2017. Our final exam will be Wednesday Dec. 6, 3:005:00PM.
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Location 
Room 114 LOCKETT Please note: this is a room reassignment and it is not the room listed in the published schedule booklet! 
Leonard Richardson 
Office 386 Lockett 
Office Hours 
MTWThF 11 AM 12 Noon. I am available at many other times. Call or email first to make sure I'm able meet with you. I answer email many times dailyusually quickly. 
Telephone 
5781568 
EMail 
rich@math.lsu.edu 
Text 
Richardson, L., Advanced Calculus: An Introduction to Linear Analysis, John Wiley & Sons, 2008. ISBN 9780470232880. There is a list of errata. If you find an error not on this list, please tell me. The text is also available as a free ebook through the LSU Library at this link. 
Graduate Assistant 
Mr. Vishnu Prasad Sivaprasad will grade those homework problems that are to be turned inthe ones that are assigned in red in the table below. He will be available to answer questions in his office, Room 282 Lockett Hall, as follows: Monday [2.50pm  3.50pm], and Wednesday [2.30pm  3.30pm]. Please be sure to write your solutions neatly and carefully so that he can read them. 
Math Major Requirements and Recommendations
Math 4031, followed by either Math 4032 or Math 4035, satisfies the Advanced Calculus requirement for the Mathematics major with a mathematics concentration. It prepares students for graduate study of mathematics and its applications. The Department strongly recommends that Mathematics majors planning graduate study in Mathematics take all three Advanced Calculus courses: Math 4031, 4032, and 4035.
Prerequisites
Both Mathematics 2057 and 2085, or the equivalent.
Attendance is required and will be counted in your final grade.
Every student's presence and participation in class is an essential part of this course. Most LSU students are conscientious and sensible about coming to class unless there is a serious, excusable reason for not being able to do so. However, there is an unfortunate tendency for some students to become discouraged as the term progresses and to cease regular attendance. This happens despite the fact that a student who is feeling discouraged has an especially great need to be in class and to ask questions. The attendance policy is intended to ensure that every student comes to class even when the going gets tough.
Attendance will be taken and recorded daily and unexcused absences will reduce your final average as follows. If UA is the number of your unexcused absences, UA/4 will be subtracted from your final average. If you need to be absent you must tell me why so that I can determine whether or not such an absence is excused. Depending on circumstances I may require documentation for your absences and documentation will always be required if you are absent from an hour test or from the Final Examination. If you are unavoidably absent on a day when homework is due, email a pdf scan file of your homework solutions directly to the grader or to me before class time.
Homework is required and will be part of your final grade
Problems, mainly proofs, will be assigned frequently. The assignments are your main work in this course. The assignments will be collected, corrected, and returned at the next class meeting. You are encouraged to seek hints to help you get started with these problems! It is required to turn in every assignment! The key to learning to prove theorems lies in how you study Advanced Calculus. It is very important to understand thoroughly how and why the proofs presented in the book and in class work. Please read the Introduction to your textbook! We will go over every collected homework problem in class, to help you prepare for tests. At the end of the course, your homework average on a 10point scale will be added to your Exam average to produce your final average.
For example, if you have no unexcused absences and if your average on the homework is 5 points out of 10, and you have an 85% exam average, your final average would be 90%. In this example the homework credit would raise your grade from B to A. This is an increase of two grade levels on LSU's +/ grading system. However, in this same example, if you had one unexcused absence, then your final average would be 89.75%, for a B+.
Proofs assigned for homework are a very important learning experience. Some students try a shortcut  copying the correct proofs from the board after the homework has been graded, without turning in their own efforts. This tends to produce proofs on tests that are written by rote from memory, and these tend to be lacking in logic and thus incoherent. It results also in low grades on Part I of each test, because the student's own conceptual errors have not been turned in and thus have not been corrected. Remember that homework is required!
In order to learn the logical structure of advanced calculus, one needs to follow a given set of definitions and theorems from start to finish. If you wish to use other definitions or theorems from a different book, you must also include a proof that the definition or theorem you have chosen is equivalent to the one we used in the course. This will require that you do much more work than is needed to follow the definitions you have been given in our course.
There are unscrupulous businesses online that will sell you solutions to homework problems. If you were to avail yourself of such a service, you would be cheating yourself out of this part of your education. The result will be an unacceptably low grade and very likely the need to repeat the course and pay tuition a second time for the same course. Your learning of Advanced Calculus will come only from your own work. There are no shortcuts. You need to turn in every assignment on time, come to class daily from the first day of the semester to the last, ask questions about everything you do not understand clearly, and ask questions about any errors indicated on your returned homework assignments.
When should you ask questions?
You should ask questions every time you do not understand something and also every time you are curious about something. Ask questions in class. If I am looking the other way and you have a questionPLEASE call out to me so I can have the opportunity to answer your question! Ask questions after class. Ask questions in my office. Ask questions by email. Please consider this: If you are approximately 20 years old, then I have been teaching this subject for approximately 2 and 1/2 times as long as you have lived thus far. So I ask you to consider that I just may be able to help you with whatever is causing you difficulty if you will permit me to do so. Please give me a chance to help you to the best of my knowledge and ability.
Lateness and Classroom Conduct
Please try to arrive on time for class. But sometimes it may be unavoidable to be late. If you are late, please come right into class, doing so as quietly as you are able so as not to disturb other students. I do not want you to wait outside in the hall. You should have as much classroom time as possible, so please just come inquietlyand take a seat even if you are late. If you arrive late and have missed the attendance sheet being passed around the room, be sure to sign it after class. Also, if homework is due that day, remember to turn it in after class.
Class time is a time for work. So when class begins please put away all cell phones, smart phones, head phones, wrist watch communicators, tablets, laptops, etc, and turn your attention to the work of the class. Thank you.
Tests
These will be closedbook tests: No books or notes are permitted, electronic, paper, or on any other medium. No internet connected or other communication or electronic devices are permitted other than a watch to check the time. Part I of each hour test will consist of a choice of 8 out of 12 short answer questions, and Part II will offer a choice of 2 out of 3 proofs. (The Final Exam will be equivalent to two hour tests.) The proofs will be modeled closely on the collected homework, and they are sometimes identical. The short questions will be small variations of homework problemsincluding those not collectedtogether with examples from the lectures and notes. Thus if you have done the homework conscientiously, you should be prepared well for all tests. If you must miss a test, it is your responsibility to speak to me as soon as possible to determine whether or not your excuse is acceptable.
Grades
There will be three hour tests, worth 100 points each, and a two hour final examination, worth 200 points. Your test average TA will be the sum of your final exam grade and your three hour test grades divided 5. Let HA denote your homework average on a 10 point scale and UA the number of unexcused absences.
Your Final Average FA will be FA = TA + HA  UA/4. Thus 10< FA <= 110. The minimum grade for each letter grade is as follows:
A+, 97
A, 93
A, 90
B+,87
B, 83
B, 80
C+, 77
C, 73
C, 70
D+, 67
D, 63
D, 60
F, below 60
You should save all your graded work for future study and in case you think your final grade is in error.
Please take note:
Number of Dropped Grades: 0
Curve: None.
General Advice
 Many students need help to learn how to write proofs. If you feel confused, it is important to see me for help as soon as possible. If you don't know how to start a homework problem, ask for a hinteither in class or in my officeor even by email. If you ask me a question about the homework, or if you email such a question to me, I may be able to think of a good hint and then I would email it to the whole class as a hint. I can guarantee you it is possible to learn to write sound proofs: Learning begins with your efforts and your persistence.
 Attend class and ask questions. Nonattendance or lax attendance is usually the first sign of impending academic difficulty. Sometimes a student feels frustrated because of not understanding the classwork. If that is the case, it is necessary to ask more questions. Advanced calculus is a subject you can learnbut you must participate in this work.
 Assignments to be turned in are collected at the beginning of class. If you arrive late, be sure to turn in your homework at the end of class and sign the attendance sheet. Do not turn it in later than that, because it is not fair to the graduate teaching assistant, who will be busy enough with the work of grading the assignments that are turned in at the proper time.
 LSU offers extensive academic support services to help students adjust to the demands of university studies: List of Frequently Used Services.
Homework Assignments and Downloads
We will update the list of assignments and tests below as the semester progresses. You will know an assignment has been updated if a duedate appropriate to this semester appears in the lefthand column. However, sometimes we will assign a problem for a certain date and then postpone it because we don't cover as much as planned in class. So check regularly for updates as to what is due and when. If you email me about a pending assignment, I may send a hint to the whole class in answer to your question, not giving your name of course! 
Academic Honesty
The University has clear policies requiring academic honesty. If you get an idea from another book or an online source, or from talking with a friend, academic honesty requires that you acknowledge your sources openly. Above all, never copy directly from another person's written work as though it were your own. Remember that your own good name is irreplaceable. This is a sound principle which will serve you well throughout your life. Moreover, on a practical level, it is very foolish claim as your own an argument from a former student in this class or from a textbook. The arguments which are copied can be recognized very easily as not coming from the student, and often the precise source can be identified readily. This means that the honorable course of action is also the practical one.

Due Date 
Assignments: Hand in problems in red for grading. The problems in red are required. Assignments must be written neatly so that the grader can read them. There is also a class of optional problems, called Bonus Problems, which are intended for those students who find the required homework easy and want to be seriously challenged. These are worth up to 20 extra homework points per problem. Bonus problems need to be turned in directly to me on a separate sheet from the regular homework, clearly marked Bonus Problems at the top. Bonus problems are due, if you choose to do one of them, one full week after the date listed, unlike normal graded homework, which is due the date listed. Bonus problems must be handed in separately from the normal homework, and they will be graded more strictly for logical rigor than the required homework. Please read the Academic Honesty policy above! 
August 21 
Read this syllabus and bring any questions you have to class today. 
August 23 
Read Pp. xxi7and try to do problems 1.11.2, 1.4. These are not to hand in, but you should write your solutions on paper in order to learn from the work. We will go over somebut not allof these problems in class according to your requests. You are responsible for asking about the questions with which you need help! You will be responsible for all assigned problems, whether collected or not. 
August 25 
Hand in these problems: 1.3, 1.7.
Also, but not to hand in: 1.5, 1.6, 1.8, 1.10, 1.11. Write your solutions on paper in order to learn from the work. We will go over somebut not allof these problems in class according to your requests. You are responsible for asking about the questions with which you need help! You will be responsible for all assigned problems, whether collected or not. 
August 28 
Hand in these problems:
1. Prove that the vector product (also called the cross product) of vectors in 3 dimensional space, does not satisfy an associative law for multiplication. (Hint: find a simple example of a cross product of 3 vectors that fails to be associative, showing the calculations.)
2. Use the result of 1.3 to prove that the real number 1 is positive. That is, prove that 1>0. (Hint: Use the Axiom of Trichotomy to see that there are only 3 potential cases for the number 1. Show that two of those cases lead to an impossibility, leaving only one possibility.) Please be sure to write your solutions neatly, so that the grader can read them! 
September 1 
Do problems 1.9. Please be sure to write your solutions neatly, so that the grader can read them! Also do: 1.12  1.16. (Remember: Only problems in RED are to be handed in!) 
September 6 
1.18, 1.201.21, 1.23  1.24, 1.26, 1.27. 
September 8 
1.19, 1.22. 
September 11 
Read Section 1.3 up to Definition 1.3.4. Do exercises 1.31, 1.33, 1.361.39. 
September 13 
1.28, 1.32. Also: 1,291.30, 1.34. 
September 15 
135, 1.421.44, 1.471.50, 1.531.55. 
September 18 
1.51, 1.57. (Hint for 1.51: Do not compute with (or even write) lim x_{n} or lim y_{n} until you have proven that these limits exist. Otherwise, you will be operating with undefined terms.) (Hints for 1.57: It may help to introduce the notations s_{n}(x) for the supremum of the nth tail T_{n}(x) of the x sequence, and similarly for the y sequence and the x+y sequence.) Also: do 1.591.61, 1.63. Please note that the BolzanoWeierstrass theorem, the Nested Intervals theorem, and the HeineBorel theorem will require your focused attention! Here is an optional bonus problem, 1.56, for those students who are looking for more challenging problems than those in the required homework. Turn this in separately from the other assignments one week from the due date printed to the left of this box, if you choose to do it. Bonus problems will be graded more strictly for logical rigor than the required homework. Remember, bonus problems will be collected separately from normal homework. 
September 20 
1.62, 1.64; Also: 1.671.70 Please note that the BolzanoWeierstrass theorem, the Nested Intervals theorem, and the HeineBorel theorem will require your focused attention! 
September 22 
1.71, 1.72.
Here is another optional bonus problem, for one week from the date to the left if you choose to do it. This question is similar to 1.70, but not identical! Suppose a_{n} is a strictly increasing sequence, b_{n} is a strictly decreasing sequence, and a_{n} < b_{n} for all n. Prove or Give a Counterexample: The intersection from 1 to infinity of (a_{n},b_{n}) is nonempty.

September 25 
Bring questions to review for Hour Test #1! Everyone is expected to have a written list of questions. Your questions will help to make the review more useful for yourself and for the whole class. Last year's first hour test is available below for you to download. Note however that what we cover in the course does vary somewhat from year to year, as does the coverage of the tests. 
September 27 
First Hour Test today. This test will cover all assignments up to this date. 
September 29 
Please download and read carefully the Hour Test #1, fall 2017, Solution Sketches and Class Statistics. 
October 2 
1.761.79; 1.80, 1.82, 1.85, 1.86. Be careful with terminology! If E is contained in the union U_{a in A}O_{a} of open sets O_{a} then it is the set {O_{a}  a is in A} that is an open cover of E. Do not confuse the open cover with its own union. The union of a family of sets is only one set! 
October 4 
1.83, 1.84. 
October 6 
1.87, 1.89, 1.90, 1.92, 1.94. 
October 9 
1.88, 1.91. Here are two more Bonus Problems, for those who seek more challenging exercises: 1.93, 1.96. If you choose to do these, turn them in separately from the regular homework a week from today. After any bonus problems are graded and returned to you, please feel free to come to my office to ask for correct solutions. 
October 11 
2.12.2, 2.42.5, 2.13. Here is another optional bonus problem, for a week from today if you choose to do it: Problem 2.16. 
October 13 
2.6, 2.7 , 2.8; (Hints:
For 2.6, use the sequential criterion to prove the limit of the function does not exist.
Do not use L'Hopital's Rule for 2.7(b)! Instead, draw a unit circle and find useful inequalities by comparing areas of triangles and a circular sector, expressible in terms of x, sin x, and tan x.) Also: 2.3, 2.14, 2.15. 
October 16 
2.192.20, 2.222.23, 2.25. 
October 18 
2.21, 2.24, 2.27. For 2.27, follow the sequence of steps given. This exercise is a theorem discovered by Cauchy. For optional bonus credit: 2.26 and 2.28. These two would be for one week from this date. 
October 23 
2.29, 2.322.34, 2.402.41, 2.432.47. 
October 25 
2.35, 2.37, 2.42. In problem 2.42, remember to prove your conclusions for each of the three stated questions! 
October 27 
Bring questions to review for the hour test. 
October 30 
Second Hour Test today. This test will cover the topics covered since Test #1. 
oCTOBER 31 
Please download Hour Test #2, Fall 2017, Solution Sketches and Class Statistics. 
November 3 
2.48, 2.50, 2.52 2.59. 
November 6 
2.49, 2.51.. 
November 8 
2.60 2.63, 2.65  2.67, 2.69. 
November 10 
2.64, 2.68 . In these exercises, you may use derivatives and L'Hopital's rule although they do not appear in this text until later. 
November 13 
2.70, , 2.73, 2,76, 2.80, 2.82, 2.84, 2.85, 2.88. 
November 15 
3.1. 
November 17 
3.3. Also: 3.2, 3.43.9, 3.11, 3.12. Here is an optional bonus problem that would be due 1 week from today: 3.14. 
November 20 
3.10, 3.25. (In 3.25, you may assume f is integrable. The strictness of the inequality is the whole problem.) Also: 3.17, 3.20, 3.22, 3.23. 
November 22 
3.30b. Only part (b) is assigned, but here is a hint anyway for the unassigned part (a): In 3.30(a) one of the directions of implication has already been proven in the text. The other direction remains to be proven. (Hint: Use theorem 3.2.4, the variant of the Darboux Integrability Criterion.) Bring questions to review for the third hour test. 
November 27 
Third Hour Test today. This test will cover assignments due before today but since the second hour test. 

Please download Hour Test #3, Spring 2017, Solution Sketches and Class Statistics. 

3.21, 3.24, 3.26. Also 3.18, 3.19, 3.27. 

3.323.34, 3.36  3.39. 

3.35, 3.40. In problem 3.40, finding a value of K and showing that it works as claimed is what is meant by showing that T is bounded. Also, 3.41  3.44. 

3.46, 3.483.51, 3.52. 

Bring questions to review for the Final Exam! Don't forget to review from the beginning of the course! This 200point exam will cover the whole course in a uniform manner, so remember to review from the beginning of the course. Your final grade for the course will be the larger of the following two:
1. The grade guaranteed by the formula provided higher on this page.
2. One letter below the final exam grade.
Thus the final exam provides a safety net that supplements the calculations specified above. 

Exam Week Office Hours: TBA 
Wed. Dec. 6, 3:005:00PM. 
Our final exam . 
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Please download Final Exam, Spring 2017, Solution Sketches and Class Statistics. 
