Welcome to the Department of Mathematics

SPRING 2017


  • Instructor: Professor Eugene Speer
    • Hill Center 520
    • 848–445–7974
    • speer at math.rutgers.edu
  • Office hours:
    • Monday 1:40–3:00 PM, Hill Center 520
    • Tuesday 10:20–11:40 PM, Hill Center 520
    • Thursday 9:00–10:00 AM, Hill Center 520
    • Or by appointment or chance in Hill Center 520 
  • Detailed course information and policies: As web page and as a PDF file .
  • Hints on writing proofs:
    • From your instructor: As web page and as a PDF file .
    • From Professor James Munkres , the author of the text used in Math 441, Introductory Topology. (The copy here was obtained from Prof. Jim Wiseman at Agnes Scott College; I do not know who wrote the prefatory remarks.)
  • Homework assignments: Click here for assignments.

Announcements and additional resources

  • 2/21/2017: The first exam is on Thursday, February 23. I will hold special office hours tomorrow, Wednesday February 22, from 2:00 PM to 3:30 PM (or somewhat later, if there is demand).

SPRING 2017


Most of the homework, workshop, and exam problems in this course will ask you to write proofs; your work will be graded carefully for both content and style. We will be talking about how to write proofs throughout the course, but here are some general guidelines.

  • Content---the logic of the proof:
    • A proof should contain a clear and rigorous chain of reasoning leading from some given hypotheses, and the definitions of the concepts involved, to the conclusion. You should give arguments supporting all the steps. (More precisely, give arguments for all the steps except for those which are ``obvious''. Learning what this means---i.e., learning how much to write down for each argument---is part of learning the art of writing proofs). Everything written down should be relevant to this chain of reasoning: don't start by writing down a list of things you know, and don't digress as you go along.
    • How does one find the chain of reasoning which makes up the proof? There are no universal answers, but here are a few hints.
      • Begin by establishing clearly in your mind what you know and what you want to prove.
      • Know and use the definitions of all the concepts involved. For many simple proofs, the logic needed is almost forced on you by the definitions you must use.
      • Use scratch paper. There you can play with various ideas and, through these, understand the logic of your proof clearly before you begin to write it up for someone else to read.
    • A common student error in writing a proof is to work backward from the desired conclusion to the given hypotheses. This may be helpful at the preliminary stage of figuring out how to construct a proof, but is never correct for the final version; such work belongs only on scratch paper. Once you have found the logic by working backwards you must rewrite the proof, moving from what you know to what you want to establish, to be sure that the logic works in that direction. Another way to say this is that in a proof you should never write down a statement or equality unless you know that it is true---from definitions or previously established results, from the hypotheses which are given, or from some chain of reasoning based on these---or unless you state explicitly that it has not been established. (For example, you might begin ``We must show that . . . ,'' or a proof by contradiction might begin ``We proceed by contradiction; suppose then that such-and-such is true.'')
    • When you are done, read your proof critically. Pretend that it was written by a stranger, and that you did not know what he or she was thinking. Does the proof then convince you absolutely? If not, try again.
  • Style---the language of the proof:
    • Write in complete, grammatically proper sentences. Remember that the equal sign is a verb. Avoid dangling modifiers.
    • Study proofs in the text, in other books, and from the lectures, to get a feeling for good mathematical style. Be aware, however, that the proofs in our text, by Abbott, are for pedagogical reasons frequently much more discursive than your own proofs should be.
    • Don't use the notations ∀ and ∃. Don't introduce mysterious abbreviations to save writing out words. You are allowed to use the abbreviation ``iff'' for ``if and only if''.
    • Read the document Writing proofs  by Professor James Munkres of MIT, the author of the text used in Math 441, Introductory Topology. (The copy here was obtained from Prof. Jim Wiseman at Agnes Scott College; I do not know who wrote the prefatory remarks.)
  • Homework and workshop write-ups---when you have time to do it right:
      • Start the homework early. Then, if you just don't know how to get started or you get lost in the middle, talk to me, to our workshop instructor Semeon Artamonov, or to other students.
      • Don't turn in scratch work. Once you have decided how to construct a proof, and written out the details once, rewrite the proof neatly to turn in.
      • When your work is returned to you, read the comments and be sure you understand their point. If you don't, come to see me or Semeon to talk about them.

SPRING 2017


  • Assignment 1: Some of the problems on the assignment sheet are marked with a star or asterisk: *. Turn in these problems, and only these problems, at the beginning of the workshop session on Wednesday, January 25.
    Assignment 1. Solutions to Assignment 1 are now posted on Sakai.

     

  • Assignment 2: Some of the problems on the assignment sheet are marked with a star or asterisk: *. Turn in these problems, and only these problems, at the beginning of the workshop session on Wednesday, February 1.
    Assignment 2. Solutions to Assignment 2 are now posted on Sakai.

     

  • Assignment 3: Some of the problems on the assignment sheet are marked with a star or asterisk: *. Turn in these problems, and only these problems, at the beginning of the workshop session on Wednesday, February 8.
    Assignment 3. Solutions to Assignment 3 are now posted on Sakai.

     

  • Assignment 4: Some of the problems on the assignment sheet are marked with a star or asterisk: *. Turn in these problems, and only these problems, at the beginning of the workshop session on Wednesday, February 15. Please note the special instructions on the assignment sheet for writing up this homework set.
    Assignment 4. Solutions to Assignment 4 are now posted on Sakai.

     

  • Assignment 5: Some of the problems on the assignment sheet are marked with a star or asterisk: *. Turn in these problems, and only these problems, at the beginning of the workshop session on Wednesday, February 22.
    Assignment 5. Solutions to Assignment 5 are now posted on Sakai.

     

  • Assignment 6: Some of the problems on the assignment sheet are marked with a star or asterisk: *. Turn in these problems, and only these problems, at the beginning of the workshop session on Wednesday, March 8.
    Assignment 6. Solutions to Assignment 6 are now posted on Sakai.

     

  • Assignment 7: Some of the problems on the assignment sheet are marked with a star or asterisk: *. Turn in these problems, and only these problems, at the beginning of the workshop session on Wednesday, March 22.
    Assignment 7. Solutions to Assignment 7 are now posted on Sakai.

     

  • Assignment 8: Some of the problems on the assignment sheet are marked with a star or asterisk: *. Turn in these problems, and only these problems, at the beginning of the workshop session on Wednesday, March 29.
    Assignment 8.

SPRING 201


  • Instructor: Professor E. Speer
    • Hill Center 520
    • 848-445-7974
    • speer at math.rutgers.edu
  • Office hours:
    • Monday 1:40–3:00 PM, Hill Center 520
    • Tuesday 10:20–11:40 PM, Hill Center 520
    • Thursday 9:00–10:00 AM, Hill Center 520
    • Or by appointment or chance in Hill Center 520
    • Or by appointment or chance in Hill 520
    • Text: Abbott, Stephen, Understanding Analysis (Second Edition). New York, Springer, 2015. We should cover the first five chapters of this text.
  • General: The primary aim of this course is to teach you the rigorous foundations of calculus---foundations which were omitted in earlier calculus courses. We will give precise definitions of the concept of a limit and of such concepts as the continuity and differentiability of functions, and will prove some of the many consequences which may be deduced from these definitions. A secondary aim is to improve your skills in reading and understanding mathematics written by others and in writing precise definitions and rigorous proofs of your own.
  • Workshops: Our Wednesday class is a workshop. Semeon Artamonov, the workshop instructor, will spend twenty-five minutes or so discussing homework, then ask you to form small groups to work cooperatively on problems. One workshop problem is to be written up and handed in the next week. Here is a discussion of how to write up the workshop and homework problems you hand in.

    Workshops are an integral part of the course; on time attendance and participation are mandatory. More than one unexcused absence from a workshop will affect your grade; you can expect to be excused for reasons of illness or when winter weather makes travel dangerous or impossible.

  • Homework: Homework problems will be assigned weekly via posting on the class web page given above. The problems are to be handed in at the beginning of the workshop.
  • Academic Integrity: You will work cooperatively on workshop problems and are encouraged to discuss homework problems with me, with Semeon, and with other students. After you have finished discussing a problem (workshop or homework), however, you must write your solution independently, not in concert with others. Your ideas and approach to the problem may come from discussion, but you should express those ideas in your own words. If you consult any source, such as a web page, you must properly acknowledge this, giving a full citation, and again, your write-up must be in your own words, not copied from a source. Failure to observe these rules will be considered a violation of Rutgers' Academic Integrity policies.
  • Exams: There will be two in-class exams, on Thursday, February 23, and Thursday, April 6; the final exam is on Monday, May 8, from 8:00 to 11:00 AM. Make-up exams will be given only in the case of well-documented illness or major emergency or (only with permission in advance) of a major outside commitment.

  • Grading: Grading will be based on a weighted average of homework, workshops, and exams:
    Homework and workshops   20%
    Class exams           20% each           40%
    Final exam   40%

     

    Back to the class web page.

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