Advanced Calculus (Math 142A, Fall 2009)


Justin Roberts

Lecture times. MWF 1-1.50, in Warren WLH 2111

My office hours are Wednesday and Friday 11-12 in room 7210, AP&M.

How to contact me. This webpage (math.ucsd.edu/~justin/142F09.html) should answer most basic questions. My email is justin@math.ucsd.edu and my phone is 534-2649.

Discussion section. The TA is Tom Laetsch, and his sections are on Tuesdays at 10 and 11 in AP&M room B412. His office hours are Tuesday 1-3 and Wednesday 2-4, in AP&M room 6452. His email is tlaetsch@math.ucsd.edu. 

The book is Victor Bryant's "Yet another introduction to analysis".  I recommended this book because it seems unusually readable, has some quite nice problems, solutions at the back, and is cheap. My lectures probably won't follow it too closely but I will be setting some homework problems from it. I will also use a supplemental free online textbook by William Trench called "Introduction to real analysis". This book is more formal in style than Bryant's (and it goes further, covering vector calculus stuff that is not part of 142AB) but it also has lots of good problems in it (without solutions) which I will be setting for the graded part of homework. Here is the link: Introduction to real analysis, by William Trench.

A rough course outline for 142A and 142B. (This isn't exactly the same as the order the books do things.)

1. The real numbers
2. Sequences
3. Series
4. Continuous functions
5. Differentiation
6. Taylor series
7. Integration
8. Sequences of functions
9. Fourier series
10. Further topics: perhaps special functions, distributions, Fourier transform...

Homework will be assigned weekly on Mondays. It will generally consist of problems from both the books above. It will be due in a "homework box" in the AP&M building at 5pm on Tuesday the week after; the discussion sections are the morning of that Tuesday, so you'll have a bit of time after section to finish things up. Probably only one of the questions will actually be graded (though there may be extra marks for completeness of homework).

Midterms There will be one in-class midterm, most likely on Wednesday 28th October. 

Final The final exam is on TBA. 

Grading policy. Homework is worth 20%, the midterm 20%, and the final 60% of the total grade. But you can discard the midterm score if you do better on the final; that means your final score is the higher of
20% homework + 20% midterm + 60% final or
  • 20% homework + 80% final.
    (This calculation will be done automatically for you at the end of term. If for some reason you miss the midterm, you will not be allowed a make-up, but automatically are covered by the second formula.)
    •

  • Homework #1, for Tuesday Oct 6
  • Do your best to write "good proofs" in the style of Math 190! The more English words, the better (well, up to a point!)
  • From Bryant: exercises p.5 nos. 1,2,3; p.8 nos 1,2
  • From Trench: exercises 1.1, no. 5
  • Also: show that between any two distinct rational numbers, there is an irrational; and that between any two distinct irrational numbers there is a rational.

    • Homework #2, for Tuesday Oct 13

    From Bryant: p.18#3,4; p.19#7; p.22#3

    From Trench: p.15#5

    Also:

    1. Give a proof of Bryant's version of the completeness axiom, starting by assuming the "least upper bound" axiom. (This, together with the proof in the other direction I gave in class, establishes that these are "equivalent" axioms.)

    2. Prove that for any positive number "epsilon", there exists a natural number "n" such that 1 over 2^n
     is less than epsilon.

    3. Show that the set of (positive and negative) integers Z is countable. Then show that the set of points in the plane whose coordinates are both integers is also a countable set.


    Homework #3, for Tuesday Oct 20

    From Bryant: p36#2, p42#1, p42#3, p43#6, p50#4

    From Trench: p192 #3, 4. You're meant to do these problems directly using the basic "epsilon" definition of convergence, not by using any "limit laws" or other shortcuts (which we will get to soon). That makes some of them quite hard, but it's very educational to try to do them this way.

    Homework #4, for Tuesday Oct 27
     
    Problems are on this sheet: 09sheet4.pdf



    Midterm will be in class on Friday Oct 30

    It will cover everything up to and including HW 4, and probably have questions similar to those on HW 4.
    Please bring a blue book!



    Homework #5, for Tuesday Nov 3
    From Bryant: p.73#1, #2, #4, #5

    From Trench: p.229#7, #8(a)(b)(d)(e), p.231#17(a)(b), #18(d)(e), #27(a). (This may look like a lot, but most of them require a quite straightforward comparison (squeezing) or application of one of the convergence tests; once you spot the trick they take only a couple of lines of writing.)

    Homework #6, for Tuesday Nov 10

    1. From Bryant: p.73#3. This question completes the analysis of when sums of powers of n converge - in class we worked out what happened when r is less than or equal to 1, or greater than or equal to 2, but not in between. It requires you to use things proved in p.19#7 and p.22#3 (which were part of HW#2).

    2. Now suppose that p(n) and q(n) are polynomials, and we look at the series "sum from n=1 to infinity of p(n)/q(n)". (Assume q(n) does not have any roots which are positive integers, so that this makes sense!). By using the comparison tests and the results of the first question on this week's HW, show that the series converges if the degree of q is greater than or equal to the degree of p plus 2, and diverges otherwise.

    3. Trench p.231#21(a)(b), p.231#22

    4. Trench p.49#4


    Homework #7, for Tuesday Nov 24 (sorry this is late - notice it's for the week after this one!)

    1. From Bryant: p.95 #1 (i),(ii); p.95 #2;  p.107 #3

    2. From Trench: p.49 #4(a),(d); p.49 #6 (a)(b)(c)(d); p.50 #20 (b)(c); p.69 #3; p.70 #10.




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