Math 4606, Fall 2009, Professor D. Hejhal This page will be used for general announcements and the homework listing. ------------------------------------------------------------------------------- LAST UPDATE: Nov 22 QUIZ ON MONDAY, NOV 23 ------------------------------------------------- ROUGH SCHEDULE OF CLASS NEXT 2 WEEKS 11-16 Kap 6-9 a bit 11-18 Kap 6-17(Wid 43), Kap 6-11,6-12(Wid 9-5) 11-20 Kap 6-12 thru 6-14, Wid 9-5,9-6 11-23 Kap 6-14 finish, review 6-15, QUIZ 11-25 review 6-16, Kap 6-10, 6-18(a bit) ----------------------------------------------------- Current Office Hours MWF 14:30-15:30 or so **** STUDENTS ARE RESPONSIBLE FOR BEING AWARE OF THE INFO GIVEN IN THE _UPDATED_COURSE SYLLABUS **** ------------------------------------------------------------------------------- Recent Lecture Emphasis: 10/12 Kap 2-23 theorem B (the completeness property of R) 10/14 Kap 2-23 theorem A, nested interval theorem, B-W thm (thm C and E), compactness property of closed + bdd sets 10/16 Kap 2-23 review of thm F, thm J (proof of max-min), intermed value theorem (problem 5) 10/19 finishing up Kap 2-23, with 'if and only if' characterization of compactness in R^k 10/21 review 10/23 exam 1 10/26 proof of implicit function theorem for F(x,y)=0 following something like Widder 56-57 10/28 part 2 of that; uniform continuity Kap 4-10 10/30 proof of Kap 4-10 (theorem 1) using simplified methodology of Wid 177-178 (alpha(x)=x); proof of Cauchy sequence property of R by B-W (see Kap 382 theorem 6) 11/2 riemann's integrability criterion; f(x) bdd, with only finitely many discontinuities is riemann-integrable 11/4 f(x) monotonic ---> riemann-integrable; basic theorems for abs(f(x)), f**2, f*g ; showing that f riemann-integrable is equiv to riemann sums converging as ||P|| ---> 0 11/6 showing that riemann-integrable is equivalent to riemann sums converging; start of Riemann-Stieltjes 11/9 physical applications of R-S integrals (Wid 5-5) 11/11 integration by parts, alpha of BV type (Wid 157+160) 11/13 review of Kap 6-1 thru 6-7 (read 6-8) 11/16 Kap 6-9 ------------------------------------------------------------------------------- HOMEWORK LISTING ================ THE STARRED PROBLEMS ARE THE HAND-IN ONES. DO *NOT* HAND IN THE UNSTARRED ONES. week 1 Kap 1-5 8(cd),10(e)(f),11,12, 15(*) Hejhal(*) Let f(x,y)= xy(2*x*x-3*y*y)/(2*x*x+3*y*y) for (x,y) not (0,0). Let f(0,0) = 0. Show that f sub xy (0,0) is not equal to f sub yx (0,0). See Widder 53. DUE ON FRIDAY SEP 18 week 2 Kap 1-9 1,9 Kap 1-15 1,4 Kap 1-16 3,4 Kap 1-17 1(abc), but 1(d)(*) in 1(d), we ASSUME that elem row operations preserve row-rank; and elem column operations preserve col-rank (see p.32 for the elem row operations) Kap 2-4 7,8,9, 12(*) Kap 2-6 4(aef); Kap 2-7 1(ade), 3(acd), 2(b)(*), 3(b)(*) DUE FRI SEP 25 week 3 Kap 2-8 4,7; 11(*) Kap 2-9 1(b), 3; 5(*) Kap 2-11 3, 4(*),7 Kap 2-12 5(*), 8(*) DUE FRIDAY OCTOBER 2 week 4 Kap 2-13 8-10,11,18, 12*(prove it!) Kap 2-18 3,7, 11; 10* Kap 2-21 12, 6e(*),6f(*), 7* DUE FRIDAY OCT 9 week 5 Kap 2-22 6(c and d)(*), 7(JUST for d)*, 10 ### For 6(d), we seek another simpler type of dependency, ie LINEAR. For that see (2.165). Cast into form (2.166) and solve! Kap 3-3 4,5 Kap 3-6 7a(*)(show work!), 7b, 9, 5 DUE FRI OCT 16 week 6 Kap 2-23 problem 5(*); you might also notice 1, 3 (N=3) DUE FRI OCT 23 (day of exam) week 7 *Carefully do problem 4 of the exam. Draw locus in color (reasonably); be sure to address the matter of endpoints and/or unboundedness. DUE FRI OCT 30 week 8 *Hejhal problem: prove the analog of Kap 382 theorem 6 in R^k, using d(P,Q) in place of |p-q| and the B-W theorem. Hint: start with a subsequence P_nj that you know converges to limit L. Play with 'd' and the triangle inequality: d(P_n,L), d(P_n,P_nj), and d(P_nj,L). Kap 4-1 9*, 10(bd)* *Hejhal problem: prove using convenient Riemann sums that the integral of x^2 dx on [0,1] is 1/3. (You may have to look up the sum of j^2 from j=1 to j=N. Or prove it yourself by induction.) DUE Fri Nov 6. week 9 Widder 153-154 2*, 18* Widder 158 6 Widder 161 3*, 4 Widder 179 5* (important problem) Kap 4-2 2(d)*, 2(a-c) DUE Fri Nov 13 week 10 Widder 165 16* Widder 169 5*, 14 (for 16*, use Riemann sums for part of it) Kap 6-7 13(a)*, 12,14 DUE FRI NOV 20 week 11 * Let s_n be a bounded sequence. let t_n = sup{s_m with m > n}. prove that lim sup s_n = lim t_n . (lim sup is defined as in Kap 380 or Wid 277-8) Kap 6-9 1(fi)*, find HOW MANY terms needed for epsilon = .01 in BOTH Kap 6-13 4*; prob 2 is very good too DUE NOV 30 =============================================================================== (UPDATED syllabus) Math 4606 Fall 2009 ADVANCED CALCULUS Instructor: Prof. Dennis Hejhal Vincent Hall 220, Phone 625-4557 Email hejhal@math.umn.edu Office Hours: _________________________________(or by app't) Textbook: W. Kaplan, Advanced Calculus, Addison-Wesley, 5th ed; Supplemental Text: D. Widder, Advanced Calculus, Dover, 2nd ed This course, for which single- and multivariable calculus are prerequisites, is intended to give students a deeper understanding of such material (i.e., of basic mathematical analysis). The course is designed to serve two types of UM students: undergraduate juniors and seniors, primarily math majors, and graduate students from outside of mathematics. Such graduate students should check that their program accepts this 4-xxx level class for graduate credit. Math 4606 is NOT designed to prepare students for 8-xxx level courses requiring an analysis prerequisite; the sequence Math 5615-5616 is designed for that purpose. The content and style of Math 4606 this semester will be a revamp of what has customarily been done over the past few years. A primary aim will be to develop material which students will find USEFUL, especially in other courses. There will be a mixture of theory, calculational aspects, and applications. Students will be expected to understand, and be able to correctly apply, the theorems that are covered. The course will (in contrast to lower-level calculus!) include some rigorous proofs. Most of the material covered will involve concepts in 1,2, or 3 dimensions. In broad perspective, the main topics in the course will be taken from chapters 1 - 5 of Kaplan, supplemented by some selection of material from chapters 6 and 7 (as determined by student input and performance). Since some of the topics are presented more succinctly or "slickly" in Widder's book, we will often use the Widder text as a supplemental reference. HOMEWORK: 'hand-in' homework will be assigned weekly and will be due the following week on Wednesday. SEE www.math.umn.edu/~hejhal/ for the ongoing master list. Unstarred problems are 'recommended' only; they should not be handed in. The hand-in HW has a (*). GRADING ALGORITHM: Hour Exam #1 100 pts Hour Exam #2 100 pts Homework (plus 2 or 3 quizzes) 100 pts Final Exam 200 pts ---------------------------------------- TOTAL 500 pts It is important to come to class since I occasionally deviate from the book in the interest of clarity (and/or streamlining) - and often develop key things by working through typical (exam-type) problems that illustrate the underlying theory in its most rudimentary form. NO make-up exams will be given. Books, notes, calculators are not allowed on the exams. Late homework will be accepted ONLY with special permission. (Exams missed due to illness or with special permission are normally made up by pro-rating the grade on the Final Exam.) SCHEDULING NOTE: the plan for the first 6 (or so) weeks of the course is roughly a) review parts of Kaplan 2.4, 2.5, 2.15, and 4.3; b) do Kaplan 1.1-1.5; then 1.6-1.9 lightly and 1.14-1.17 lightly; c) Kaplan 2.1-2.12 (special emphasis on 2.10 and 2.12!!); 2.13-2.20 lightly d) Kaplan 3.1-3.6; e) shift gears (!) -- do Kaplan 2.23, 4.10, and Widder Ch 5 Sec 6. FURTHER NOTES. i) WITHDRAWALS: see _Fall Semester 2009 Class Schedules_ for policies concerning withdrawals. ii) SCHOLASTIC CONDUCT: see www.it.umn.edu/students/policies/index.html iii) INCOMPLETES can be given only in accordance with Math Dept guidelines (in particular, only _prior to_ the Final Exam, and with a previous record of satisfactory work).