General information
| Schedule |
Mon, Wed, Fri, 1:15 - 2:05 |
| Location | Building 380 (Math), room 380-D
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| Professor |
Brian Munson (munson@math.stanford.edu)
Office: 382-H (2nd floor, Math building 380)
Office Phone: 3-7829
Office Hours: Monday 2-3:30, Tuesday 1-2:30, or by appointment.
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Course
Assistant |
Andres Angel (andresangel@stanford.edu)
Office: 380-T
Office hours: Monday 4-6, Tuesday 11-1 & 3-5, Thursday 11-12.
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| Textbook |
Elementary Analysis: The Theory of Calculus, by Kenneth A. Ross. |
Course
content | We will cover most of Ross' book. We will begin with a discussion of logic and proof techniques, and then move on to talk briefly about the number systems. Then we'll talk about sequences, our major tool for studying continuous functions. After this we will talk about differentiation, integration, and, if there is time, sequences and series of functions. Please see the syllabus for a more detailed description. |
| Prerequisites |
The only prerequisite for this course is Math 51, which deals with linear algebra and some multivariable calculus. In theory, you could take this course having only taken single variable calculus, but the more advanced and abstract mathematics you've been exposed to, the easier this course will be for you (that's the theory, at least). However, I'm working under the assumption that this is the first proof-based course you've taken. |
Announcements & Dates
- Important Dates
- October 12th: Add deadline
- October 15th: Exam 1
- October 19th: Drop deadline
- November 12th: Exam 2
- December 8th: Final Exam, 8:30-11:30 AM, Room 380-C
- Guidelines for the literature reaction papers. Discussion and due dates are listed in the homework section of this page.
- Solutions to Exam 1
- Exam 1 will cover through section 10 of the book. You are not responsible for section 6. As for as the discussion of proofs done the first few days in class, you are responsible for this only in that you need to know how to apply it to prove particular statements such as those in our text. The point of those first few days was to give you some proof tools to work with.
- Exam 2 will cover through section 28 of the book, with an emphasis on sections 11 through 28. Keep in mind that we skipped sections 13, 16, 19, 21, 22, and all of chapter 4. The list of topics is basically this: limsup and liminf, series, continuous functions, and (the beginnings of) differentiable functions. The format will be very similar to the last exam. You will be asked to state several definitions, prove various statements, and decide whether certain statements are true or false. You will also be asked to state and prove exactly one of the following theorems from the book, to be chosen at random. They are: Theorem 12.2 (I want you just to know how to prove the inequality of limsups, just as the proof in the book.), Theorem 14.9 (The Root test), Theorem 17.2 (equivalence of sequential continuity and ε-δ continuity), and Theorem 18.2 (The Intermediate Value Theorem). The proofs of these theorems illustrate important ideas and techniques in analysis, and in order to memorize them you're going to need to understand them. This should additionally help you hone your analysis skills.
- Solutions to Exam 2
- A little note I wrote for you to see when and how p-series diverge and converge without using the integral test.
- The Final Exam will be held at the time and date listed above. Note the different room from where we usually meet! It will be comprehensive, although it will favor the material covered since the second midterm (differentiation and integration). The format will be similar to that of past exams. There will be definitions, true-false questions, and I will ask you to prove various statements. You will not be asked to memorize the proofs of any theorems for the final. Although I think this is in general a good idea, there are several compelling reasons not to. One is that our final is on the first day at the earliest possible time, and you wouldn't have enough time to do so (thank you for pointing this out: I forgot and wouldn't have remembered until it was too late). Another reason is that you're going to need time to study the integration stuff that we've been learning recently. Yet another reason is that, well, you have other finals (something I also forgot, in some sense). Perhaps the best reason is that I can find a way to test you on the same material without asking you to memorize proofs. I hope to write the exam so it will take you a little more than 2 hours to get through it, with the rest of the time for polishing your proofs and working on the more challenging problems. I'm thinking there are going to be about 10 to 12 questions, each of similar length to a regular exam problem. Now here are some specifics on the sections we covered in this class, which consists of a brief list of topics and the stuff from certain sections you're not responsible for (since we didn't cover everything).
- Sections 1, 3 (the properties of the number systems)
- Section 4-5 (ideas of sup and inf)
- Section 7-12 (sequences, limsup & liminf)
- Section 14 (series), Section 15 (only the Alternating Series Test)
- Section 17 (continuous functions), Section 18 (properties of continuous functions, don't worry about theorem 18.4 and beyond)
- Section 20 (limits of functions)
- Section 28 (differentiation), Section 29 (properties of differentiable functions, don't worry about 29.9 and beyond)
- Section 32 (integration, don't worry about 32.6 and beyond)
- Section 33 (properties of integrable functions, don't worry about knowing what a piecewise continuous and piecewise monotonic function are. You already know what a continuous function and a monotonic function are, and that's enough.)
- Section 34 (Fundamental Theorem of Calculus. Don't worry about example 3. Although I didn't talk about integration by parts or change of variable, you should know how to use them. I'm assuming you've seen this in a single variable calculus course.)
Homework
Homework assignments are due in class one week from the day they are assigned, unless otherwise noted. Homeworks are typically due Wednesday, and the homework due on a given Wednesday covers material from the previous week. Homework will be assigned on a weekly basis.
Solutions
|
Assignment |
Due date |
Relevant Reading |
| #1 |
Handout
| Due: Wednesday, October 1 |
| #2 |
Chapter 1:
- 1.4, 1.9, 1.11
- 3.6, 3.8
- 4.(1-4)aehinrw, 4.5, 4.6, 4.14
- 5.3, 5.6
| Due: Wednesday, October 8
|
- Monday 9-29: 1.1 & Appendix on Set Notation
- Wednesday: 1.2 & 1.3
- Friday: 1.4 & 1.5
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| #3 |
Chapter 2:
- 7.1, 7.4, 7.5b
- 8.2ab, 8.4, 8.5, 8.8b
- 9.2, 9.4, 9.10, 9.18
- 10.6
- 11.8a
- 12.4
| Due: Friday, October 17
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- Monday 10-6: 2.7 & 2.8
- Wednesday: 2.9
- Friday: 2.10
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| #4 |
Chapter 2:
- 11.2, 11.8b
- 12.2, 12.3, 12.7, 12.8, 12.13
Due: Wednesday, October 22 |
- Monday 10-13: 2.11
- Wednesday: Exam 1
- Friday: 2.12
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| #5 |
Chapter 2:
- 12.14
- 14.1bef, 14.2abde, 14.5, 14.6, 14.12, 14.14
- 15.1, 15.6, 15.7
Due: Wednesday, October 29 |
- Monday 10-20: 2.14
- Wednesday: 2.15 (Only Theorem 15.3)
- Friday: Literature Discussion, 1 page reaction due.
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| #6 |
Chapter 3:
- 17.1, 17.4, 17.8, 17.10abc, 17.12, 17.13
- 18.1, 18.6, 18.8, 18.9, 18.10, 18.12
Due: Wednesday, November 5 |
- Monday 10-27: 2.15 & 3.17
- Wednesday: 3.17
- Friday: 3.18 (be aware of, but don't worry too much about theorems 18.4 and 18.5. The main thing to understand from this chapter are theorems 18.1 and 18.2.)
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| #7 |
Chapters 3 & 5:
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| #8 |
Chapter 5:
- 29.1, 29.2, 29.3, 29.5, 29.10, 29.11, 29.14, 29.18
Due: Wednesday, November 19 |
- Monday 11-10: 5.29
- Wednesday: Exam 2
- Friday: 5.29
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| #9 |
Chapter 6:
- 32.1, 32.2, 32.5, 32.6, 32.7
Due: Wednesday, November 26 |
- Monday 11-17: 6.32
- Wednesday: 6.32
- Friday: Literature Discussion, 1 page reaction due.
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| #10 |
Chapter 6:
- 33.2, 33.4, 33.7, 33.8, 33.10, 33.13, 33.14
Due: Wednesday, December 3 |
- Monday 11-24: 6.33
- Wednesday: 6.33
- Friday: Food hangover from Thanksgiving. No class.
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| #11 (Suggested problems, not to be handed in) |
Chapter ?:
- 34.1, 34.2, 34.3, 34.5, 34.6, 34.11, 34.12
- I think I'm going to suggest you start studying.
- Look at your old homeworks and solutions on the web.
- NOTHING BESIDES THE FUNDAMENTAL THEOREM OF CALCULUS FROM THIS WEEK WILL BE ON THE FINAL EXAM.
Due: Not due. |
- Monday 12-1: 6.34
- Wednesday: Maybe "review", where I hit on the main themes of this course and how they arise. I'll also try to mention the literature at various points and what it had to say about these themes.
- Friday: Maybe I'll give a lecture on some toy problem that we can tackle using our tools, or maybe I'll talk about "the rest of math" as I know it. This means it will be skewed towards things I know.
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