The general structure of each week will be as follows: on Monday, you'll turn in homework based on the last week's lectures; each day will start with a short (5 min.) quiz asking you to recite some definitions or statements of theorems from the previous day's lecture (I'll give guidance on anything you're "not responsible for"); each day some of you will be called on to present some problems from the book on the board (I will warn you which in advance); on Thursdays we will have a quiz of moderate length (25 mins?) in which you have to reconstruct an important proof from that week's lectures (I will start off giving you lots of help here and we will build up to the point where you don't need any). Of course, every day I will also be lecturing on material. I also hope to provide time for you to work in groups on problems during class time.
There will be three exceptions to this: the first week there will be no homework due on the Monday (as that's our first class meeting), nor will you be expected to present anything on the board that day as I won't have been able to warn you all; the fourth week we have the midterm, so I will cut some of the other assessments to allow us time to prepare; the eighth week we have the final and we will also be doing a topic (metric spaces) which is not in the book, so things will work a little differently.
At some point (ie. when I've planned in more detail), this will become "day by day". Update 6/19. I've added supplemental reading to most weeks.
Week 1. Axioms for number systems; sequences. Burn: chapters 1, 2, 3 (-.47) [Supp: Oxford I Axioms sheet, lecture notes sections 1, 2, 5.1-.4, 7.1-.3, 8.1]. This week, I will introduce analysis by looking at the idea of approaching math via a system of axioms. We won't quite be taking an axiomatic approach in this class, but it's important to be aware of what it would mean if we did. It's also a convenient way to review mathematical induction and properties of inequalities and to provide a good way in to mathematical proofs for people who are less confident with them. We will also start studying sequences.
Day by day: M Burn Chap 1; Tu Oxford Axioms Sheet, Oxford I (1,2), Burn 2:1-29; W Burn 2:30-66, 3:1-16 [2: 30-32, 61-63. 3: 1, 4, 5, 8, 9]; Th Burn 3:17-47 [17, 20, 27]. The questions in [square brackets] are the ones you should be prepared to present to the class.
Week 2. Completeness. Burn: chapters 3 (remainder), 4 [Supp: Oxford I 3, 5-9]. Last week, I left you in suspense by not telling you one of the axioms for the real numbers. This week, we'll develop the requisite understanding to appreciate it (the axiom of completeness). We'll do this by looking at the convergence of sequences. Note: we'll lose a day for the fourth of July holiday this week.
Day by day: M Burn 3:48-4:21 [3:49, 54 (i)-(v), 68]; T 4:22-:54 [24, 33, 37, 52]; W 4th of July, no meeting, Happy Holiday!; Th 4:55-:82 [56, 62, 82]. (We're skipping the last two questions, and we'll come back to that material when we need it at the end of week 3). As before, the questions in [square brackets] are the ones you should be prepared to present to the class.
Week 3. Series. Burn: chapter 5 [Supp: Oxford I 10, 11, 13, 14]. This week, we'll apply our knowledge of sequences to investigating infinite sums.
Day by day: M Burn 5:1-30 [3, 4, 19, 27]; T 5:31-55 [33, 44, 45, 51]; W 5:62-90 [79]; Th 6:91-113 [97, 101].
Week 4. Midterm; Functions and continuity Burn: 6,7 [Supp: Oxford II 1]. We'll start the week with a midterm on everything we've learnt so far (chapters 1-5, plus stuff on axiomatics not in Burn). I hope this will be on Tuesday, but that does rely on me getting a single room for the two hours, as I don't want to move rooms halfway through an exam. If not, it will be on Monday. After the midterm, we will start looking at functions instead of sequences and the notion of continuity. We will see how some consequences of continuity that seem obvious are actually subtle and tightly bound up with our work on completeness.
Day by day: M Burn 6:1-55 [5, 15,16, 27, 49]; T 6:56 - 7:7 [63, 80]; W Trip to Disneyland (or Midterm... tbc); Th 7:8-48 [9].
Week 5. Differentiation. Burn: 8,9 [Oxford II 2]. We will develop the machinery of derivates and then look at what theorems involving derivatives completeness gives us. We may also be finishing up chapter 7 this week and so have to delay some of this material to next week.
Week 6. Integration. Burn: 10 [Rudin Chap 6 (through thm 22), Binmore Chap 13]. We will look at the question of when it makes sense to calculate the "area under" a function. In your calculus classes, you saw what this meant for continuous functions on bounded intervals and maybe a few mild generalizations and spent much time calculating the areas. Here, we're less interested in calculating the areas than asking when the question even makes sense. In your second course on analysis you will study a better answer to this question (the Lebesgue integral), but here we will study an integral which we can rigorously develop in the time available and is pretty powerful (the Riemann integral). If necessary, we may be finishing off chapter 9 this week. If not, we'll either look at some different ways to develop the exponential, logarithmic and trigonometric functions (one example is Burn's chapter 11, but we'll also survey others), or review some topics of the class's choosing.
Week 7. Uniformity Burn: 12. We'll look at the notion of uniform convergence. This is an important topic in its own right (and in terms of applications of analysis and prominence in further study of analysis), and also will serve as a review of everything we've done since week 5, as it interacts with pretty much everything. If there's time, and we didn't do it last week, we'll do the side lecture on exponential, etc., functions here. Or, I could say something of interest to any applied mathematicians in the class about differential equations. Your choice.
Week 8. Metric spaces; final. In the last week, we'll talk about metric spaces. This notion is basic for much further work in analysis and applications (especially to mathematical physics). It will also give us the equipment to very quickly deduce all the theorems we need about IR^n from the corresponding theorems we've already proved about IR. There won't be time to have a formal homework due, so as an encouragement to study them well on your own, I promise to put a very easy question on the final about them. The course will end with the final.