Math 411 0201, Fall 2014 TuTh 12:30pm - 1:45pm MTH 0303 Dr. M. Machedon. Office: Math 3311. Email mxm@math.umd.edu Office Hour: Tuesdays and Thursdays 2-3 Textbook: Advanced Calculus, Second Edition, by P.M. Fitzpatrick, ISBN 0-534-37603-7 Course description: We will cover Chapters 10-17 of Advanced Calculus by P.M. Fitzpatrick. We will also go quickly over some of the topics in chapters 18-20, but these will not be covered on the final exam. The prerequisite for this class in Math 410. Grading: Homework = 20%, 2 In Class Exams= 25% each, Final Exam= 30%. Students who get less than 50% of the maximum possible score will receive an F. The two in-class exams will be on Thursday, October 16 and Thursday, November 13 postponed to Thursday, November 20. The final exam will be on Friday, December 19 1:30-3:30pm Make-up policy: There will be no make-ups for in-class exams. In the case of an absence due to illness, religious observance, participation in a University activity at the request of University authorities, or other compelling circumstances, your blank grade will be replaced by the average of your other in-class exam and the final exam, weighted equally. No late homework will be accepted. Homework assignments missed due to one of the above reasons will be replaced by the average of the other homework grades. The major grading events for this class are the two in-class exams and the final. I will accept a self-signed note which acknowledges valid reasons for missing one exam, but will require formal written documentation (such as from a medical provider) for subsequent absences. After each in-class exam students have two weeks to appeal the grading. Appeals for the final grade must be made in writing. During exams, students are expected to apply the ideas they learn to some problems that are significantly different from the examples and homework they have seen. 1
2 On exams students must write by hand and sign the following pledge: I pledge on my honor that I have not given or received any unauthorized assistance on this examination. This does not apply to homework, where it is acceptable to exchange ideas with other people. Students who require special examination conditions must register with the office of the Disabled Students Services (DSS) in Shoemaker Hall. Documentation must be provided to the instructor. Proper forms must be filled and provided to the instructor before every exam. The Universitys policy on religious observance and classroom and tests states that students should not be penalized for participation in religious observances. Students are responsible for notifying the instructor of projected absences within the first two weeks of the semester. This is especially important for final examinations. I will communicate with the class by e-mail. You are expected to have a correct e-mail address. You can update your e-mail address at http://www.testudo.umd.edu/apps/saddr/ All problem sets are due at the beginning of class, as follows: Problem set 1, due Thursday, September 11 10.1: 2, 4, 6 10.2: 1, 2, 5 10.3: 3, 4 Problem set 2, due Thursday, September 18 11.1: 4, 11 11.2: 6, 7 11.3: 2, 9 11.4: 1, 4, 5 Problem set 3, due Tuesday, September 30. 12.1: 2a, 3, 4 12.2: 1, 6 12.3: 2a Problem set 4, due Tuesday, October 7 12.4: 1, 2, 10 12.5: 1, 2, 3 The first in-class exam will be on Thursday, October 16. It will cover Chapters 10, 11, 12,
1. Notes on compact sets. This is similar to ideas you learned in Math 410, except open sets had not yet been defined. Definition 1.1. K R n is compact if for every covering of K by open sets V α there exists a finite subcover V α1, V αm (that is, if V α are open sets so that K α V α then you know for sure there exist finitely many V α such that K V α1 V αn ) The following will be checked in detail in class: If K is compact, then K is bounded. (Use V m = B m (0) ). If K is compact, then K is closed. (Proof: Let u i a sequence in K converging to u. If u is not in K, then the sets {1/m < x u < m} are an open cover of K. This can t have a finite subcover (why?). Conversely, we prove that K closed and bounded implies K compact. Proposition 1.2. If K is a closed subset of a compact set L, then K is compact. Proof. Let {V α } be an open cover of K. Then {V α } together with the open set K c cover L. There exists a finite subcover of L consisting of some V α s and possibly K c. That finite subcover is also a cover of K, and you can remove K c from it, since it has no points in common with K. Now let K be closed and bounded. It is contained in some closed cube C, so it suffices to show a closed cube is compact. Proposition 1.3. Let C be a closed cube in R n. Then C is compact. Proof. Let {V α } be an open cover of C. Assume, by contradiction, it has no finite subcover. Divide C into 2 n subcubes (of side half the side of C). At least one of them say C 1, has no finite subcover of V α s. Divide it again into 2 n subcubes. At least one of those cubes, called C 2 has no finite subcover. Repeat the procedure to get a nested sequence...c 3 C 2 C 1 C of cubes with sides converging to 0, and the property that none can be covered by finitely many V α s. By the nested cubes theorem (exercise) there exists a unique u C i for all i, and that u belongs to at least one V α0. But then for all i sufficiently large, C i is covered by just one V α0, contradiction. 3
4 Practice problems for exam 1. These problems are meant to help you study for the exam. You can discuss these with other students. I will solve some in class, and and will give hints for others. 10.1: 7 11.1: 6 11.4: 3 12.1: 3 12.2: 14 12.3:8 12.4:10 12.5:6 1) Let f : R R continuous. Prove that the graph G = {(x, f(x)) x R} is closed. 2) Is the converse to 1) true? 3) Let f : [0, 1] R continuous. Prove that the graph G = {(x, f(x)) x [0, 1]} is sequentially compact. 4) Is the converse to 3) true? Problem set 5, due Tuesday, October 28. 13.1: 1 13.2: 4, 6, 13 13.3: 6, 11 Problem set 6, due Tuesday, November 4. 14.1: 11 14.2: 1, 10 14.3: 5, 7 Problem set 7, due Tuesday, November 11. 15.1: 5 15.2: 2, 4, 8 15.3: 1, 5 Practice problems for exam 2, not to be turned in: 13.1: 1, 8 13.2: 6 13.3:11 14.2: 1, 3 14.3: 7 15.2: 8 15.3: 3
5 Problem set 8, due Thursday, December 4 16.1 : 5, 13, 14 16.2: 2, 5, 8 16.3 : 3, 11 Recommended problems for Thursday, December 11 (not to be turned in) 17.1: 7 17.2: 1 17.3: 1a, 5 17.4:2, 12 The final exam will cover all the course material, with special emphasis on chapters 16 and 17. Review problems for the final exam: 1) Let F : R 3 R 2 be C 1. Assume F (0, 0, 0) = (0, 0) and the derivative matrix equals ( ) 0 0 1 DF (0, 0, 0) = 1 0 0 a, ) Circle the correct statement (no proof required). There exists r > 0 and g : ( r, r) R, h : ( r, r) R, continuously differentiable, such that g(0) = 0, h(0) = 0 and F (x, g(x), h(x)) = (0, 0) for all x < r F (g(y), y, h(y)) = (0, 0) for all y < r F (g(z), h(z), z) = (0, 0) for all z < r (b, ) Are the other two statements, which aren t true for sure, possible with g, h continuously differentiable and DF (0, 0, 0) as above)? An example or proof they are impossible is required. 2) Let f 1, f 2 : R 2 R of class C 2. Consider the zero sets Z 1 of f 1, Z 2 of f 2. Thus Z 1 = {(x, y) f 1 (x, y) = 0}, and the same for Z 2. Assume f i (x, y) 0 if (x, y) Z i, and assume there exists a function λ : R 2 R such that f 1 (x, y) = λ(x, y) f 2 (x, y) for all (x, y) R 2. Assume the intersection of Z 1 and Z 2 is non-empty. Prove that Z 1 Z 2 contains infinitely many points. 3) Let X = (2, ) with the the usual metric d(x, y) = x y. (a, 10 pts) Define what it means for T : X X to be a contraction.
6 (b, 10 pts) Prove or disprove: If T : X X is a contraction, then there is at most one x X such that T (x) = x. (X as above). (c, 10 pts) Prove or disprove: If T : X X is a contraction (X as above), then there exists at least one x X such that T (x) = x.