MAT321: Analysis II Spring 2012
Course Description: 4 hours, 4 credits. Continuation of MAT 320. Infinite series and power series, pointwise and uniform convergence, n dimensional Euclidean space, metric spaces, functions from Rn to Rm, continuity, and the differential as a linear map: inverse and implicit function theorems. This course is required for admission to many graduate school programs including masters in mathematics.
Official Syllabus: linked here
Instructor: Professor Sormani is a metric geometer and a differential geometer.
Office: Gillet Hall 200B, Email: sormanic at member.ams.org
Office hours for this class will be a half hour immediately preceding each meeting 5:30-6:00pm and a half hour after each meeting 7:40-8:10 pm..
Prerequisites: Vector Calculus, Linear Algebra and Analysis I
Textbook: Folland's Advanced Calculus (available used for $43)
ISBN-10: 0130652652 | ISBN-13: 978-0130652652
There are other books covering the same subject matter that students may wish to consult.
Grading: This course will have 10 Quizzes (4 points each), 2 Midterms (20 points each) and a Final Exam (20 points)
totaling to 100 points.
Location and Meeting Times: Gillet Hall 227, 6:00-7:40 pm Mon/Wed (tentative)
Tentative Schedule: (28 lessons, this website will be updated with exact dates as well as homework)
Mon 1/28: § 1.1-1.2: Introduction to Metric Spaces and Euclidean Spaces
(See also handout from Marsden: Elementary Classical Analysis)
Homework: Read about Metric Spaces, balls and Open sets.
1)a) Prove (-infinity, b) is open in R. 1)b) Prove (a,b) is open in R
2)a) Prove {(x,y): x>5 } is open in R^2. 2)b) Prove {(x,y): x>0 and y>0}is open in R^2
3)a) Prove that U and V open implies U union V is open in any metric space.
3)b) Prove that U and V is open implies U intersection V is open in any metric space.
Wed 1/30: § 1.1-1.2: Open sets in Metric Spaces and Euclidean Spaces
Classwork on Proving: (a) prove the countable union of open sets is open, (b) prove an annulua is open, (c) prove the x axis in the Euclidean plane is not open, (d) Prove {(x_1,x_2): x_2>x_1} is open in the Euclidean plane. (e) Prove {(x_1,x_2,x_2): 2x_1+3x_2+5x_3>30} is open in Euclidean space. Also in class we defined a limit is a metric space and proved it was equivalent to the topological definition of a limit.
Homework: spend four hours on the homework and be sure to do 1a, 2a, 3 and 4: 1)a) any line in the Euclidean plane is not open, 1b) any plane in Euclidean three space is not open. 2) Let X be a metric space, x,z in X and r>0 Prove: a) U(x,r)={ y: d(x,y)>r} is open and b) U(x,z,r)={y: d(x,y)+d(y,z)<r} is open. 3) Let U be an open set, so we know for all x in U there exists r_x>0 such that D(x, r_x) subset U. Show U= union_{over x in U} D(x,r_x). This is a huge union of possibly infinitely many balls and yet it is still inside U. This proof has two directions. 4) Let K be a subset of a metric space X, and r>0, we define the tubular neighborhood about K or radius r denoted T_r(K)={x: exists y in K such that d(x,y)<r}. Prove that T_r(K) is an open set. 5) Complete classwork problems d and e looking up the formulas for a distance to a plane or a distance to a line as needed.
Mon 2/04: § 1.3: Continuity in Metric Spaces and Euclidean Spaces
(See also handout from Marsden: Elementary Classical Analysis, Section 4.1)
Quiz I on Open Sets postponed due to flu epidemic
Homework: (1) Let x be a point in a metric space X and define f: X to R as f(y)=d(x,y). Prove that f is continuous. (2) Prove that if f_1 and f_2 are continuous functions from X to R then their sum is a continuous function. (3)(a) Let x,z in a metric space X and define f: X to R by f(y)=d(x,y)+d(y,z). Prove f is continuous. (3)(b) Describe the preimage f^{-1}(-\infty,r)={y: f(y)\in (-infty,r)}. It should be familiar from the last homework set. (4) Prove (2)(b) from the last homework set using (3) and the preimages of open sets are open theorem.(5) Practice for the Quiz by redoing problems from the first homework assignment.
Wed 2/06:§ 1.4: Limits, Continuity, and Sequences in Metric Spaces and Euclidean Spaces
Quiz I on open sets today.
(See also handout from Marsden: Elementary Classical Analysis)
Homework: (1) Prove [a,infty) is closed. (2) Prove a plane in Euclidean space is closed. (3) Prove the closure of a ball about x of radius r in Euclidean space is {p: d(x,p) less than or equal to r} (4) Prove that if U is an open set in a metric space X then X\U is closed.
Mon 2/11: § 1.5: Sequential Compactness
(See also handout from Marsden: Elementary Classical Analysis, review Analysis I material like sup, inf, Cauchy sequences and Bolzano Weierstrass Theorem)
Homework: (1) (a) let diam(K) = sup { d(x,y): x\in K and y \in K}. Prove the diameter is finite when K is sequentially compact by finding an upper bound for this supremum. (1)(b) Show there exists x_i and y_i in K such that diam(K)= lim d(x_i, y_i) using the properties of supremum. (1)(c) Show that when K is compact there exists x,y in K such that diam(K)=d(x,y). (2) Prove the lemma: If f: X \to Y is continuous and lim x_i = x then lim f(x_i) = f(x) using definitions of continuity and of limit. (3) Using any theorem from today as you wish, prove that if f: [0,1]^3 \to R is continuous then it has a maximum and a minimum. (4)(a) Prove that l_2 ={(x_1,x_2,x_3,....): infinite sum x_i^2 < infinity} is a metric space (4)(b) Prove that If a sequence in l_2 converges then its components converge (4)(c) Prove that the sequence p_1=(1,0,0,0...) p_2=(0,1,0,0..) p_3=(0,0,1,0,0,...) has no converging subsequence using a proof by contradiction as described quickly in class. Information about scholarships for math and comp sci majors
Wed 2/13: § 1.6: Compactness
(See also handout from Marsden: Elementary Classical Analysis)
Homework: (1) Prove that if A is a bounded set and then for any x, d(x,A)=inf{d(x,a): a in A} is finite. (2) Prove that the cover [0,1] subset union (1/alpha,2) has no finite subcover: "no matter how we choose alpha_1,... alpha_N, ..." (3) Prove that compact implies sequentially compact imitating the proof that closed and bounded sets in R^n are compact but replacing the rectangular blocks with balls as explained in class. Quiz II on Limits of Sequences and Continuity is due 2/20 and is the four problems assigned in class on Monday 2/11.
Wed 2/20: § 1.7: Completeness (See also handout from Marsden: Elementary Classical Analysis)
Homework: (1) Prove that the rationals with the distance d(x,y)=|x-y| is not a complete metric space. (2) Write in complete detail the proof that l_2 is a metric space and that it is a normed linear space and that it is complete. Quiz III on Continuity (due 2/25): QIII(a): Prove the f(x)= x^3 is a continuous function on the line. QIII(b):Prove that F: R \to R^3 is continuous iff every component is continuous (be very explicit and careful with your choices of delta depending upon epsilon) QIII(c): Prove that x^2+4y^2<25 is open in R^2 using the preimage of open is open theorem. QIII(d): A Lipschitz function f: X to Y is a function such that Lip(f)=sup (d_Y(f(p), f(q)) /d_X(p,q) ) < infinity. Prove that a Lipschitz function is continuous. Email me if you would like a copy of solutions to Quiz II emailed to you.
Mon 2/25: § 1.8: C[(0,1)], Completeness of C[(0,1)] (see also Marsden)
Homework: Try Quiz III (d) again: the point of this question is to assume the function is Lipschitz so there is some L=Lip(f) and then do a proof of continuity. Also do the Practice Midterm Exam .
Wed 2/27: Arzela Ascoli Theorem and Contraction Maps on Metric Spaces (See handout from Marsden)
Quiz IV will be due on 3/06
Mon 3/04: Midterm I: This midterm will cover proofs of open and closed sets, continuity, limits, similar to The practice exam as well as short answer questions on all concepts. After taking the midterm, review your Vector Calculus particularly Partial Differentiation.
Wed 3/06: § 2.1-2.2: Review of Differentiability
HW: 2.1/1,2,6,8,9; 2.2/1,2,7,8 Quiz IV on Derivatives will be in class on 3/13
Mon 3/11: §2.3-2.4, 2.10: Vector valued functions and the Multivariable Chain Rule
HW: 2.3/7, 2.10/1,5,7,8,9 due on Monday 3/18
Wed 3/13*: In class Quiz on 2.1-2.4.
Review your Linear Algebra particularly matrices, kernels, domain, range,and determinants before Monday.
Mon 3/18: § 2.6- 2.7: Taylor's Theorem (multiple variables)
Complete the derivation of Taylor's Theorem as described in class.
Wed 3/20: § 2.8-9: Critical Points, Extremal Values and Lagrange Multipliers
Homework: 2.6/ 1a, 2, 2.8/1a,e, 5; 2.9/4, 9, 10 18, 19. Due Wed 4/03. Quiz V: a) Find the Taylor series for f(x,y)=(x+y)^n and show eventually the sum ends and has no remainder. b) Find the Taylor series for f(x,y)=e^x sin(y) around (1,pi) (write it out including all the third partial derivatives).
Spring Break Review your Linear Algebra particularly matrices, kernels, domain, range and catch up on old material,
Wed 4/03: § 3.1-3.2: The Implicit Function Theorem and Reading a Textbook
Homework: 3.1/1, 3, 6; Read 3.2 carefully (as taught in class); 3.2/1a, 2abc,3a, (try all problems!!!)
Mon 4/08: § 3.2-3.3: Contraction Mapping Principal and a constructive proof of the Implicit Function Theorem, also Smooth Curves and Parametrized Surfaces, see Ratzkin's Notes
Homework: 3.2/1dg, 3d, 4; 3.3/2a, 4a, 5ab [Please return all graded work with resubmissions!]
Wed 4/10: Applying the Constructive proof of the Implicit Function Theorem [Please return all graded work with resubmissions of Quizes I-IV.] No resubmission for Quiz V allowed, but the following extra credit due Mon 4/15 for those who wish to improve their Quiz V grade: Find the Taylor Series for (x+5y)^4 about (2,3). The final deadline for resubmission of Quiz IV is Wed 4/17.
Mon 4/15: § 3.4: Quiz VI and Transformations, Coordinate systems and the Inverse Function Theorem
In class Quiz VI on applying the Implicit Function Theorem and Reading a Text.
Wed 4/17: § 1.7: Connected Sets
Quiz VII: 1) a) Prove [a,b] is connected b) Prove the Real line is connected 2) Prove {(x,y): x^2+y^2=1} in R^2 is connected (hint: use continuous image of connected sets are connected theorem). due 4/24
Mon 4/22: Review of Integration, Jacobians and surface areas using your own Calculus Textbook
Homework: practice using Jacobians and taking surface integrals using your calculus textbook.
Wed 4/24: Divergence Theorems and Green's Thm in 2D (§ 5.1-5.4)
Homework: practice using 2D Divergence and Green's Theorem using your calculus textbook. Quiz VIII: apply three dim Divergence Thm to prove the integrals relating Laplacians and gradiants of functions due Mon 4/29
Mon 4/29: Flux Integrals and 3D Divergence Theorem
In class Quiz IX: on reading a text and applying the 3D divergence theorem to a specific vector field and region due Wed 5/01. [If you did not do Quiz VIII then submit it Wed 5/08]. Homework (do before Midterm II!): Handout from Larsden Textbook: 1118/ 15, 17, 19, 23, 25; 1126/1,9,21,27,28
Wed 5/01: Review for Midterm II
Mon 5/06: Midterm II: This midterm will cover all material since Midterm I including material similar to Quizzes IV-VIII. In particular: 1) verify a given function is differentiable at a given point, 2) Find the Taylor expansion of a given function about a given point up to 2nd derivatives, 3) Given a function determine if is is locally invertible near a given point, 4) Find the area of a surface and perform a change of variables in an area integral using a Jacobian, 5) Determine if a 2D vector field is conservative and find its potential, 6) Evaluate a Flux Integral over a given surface.
Wed 5/08: Conservative Vector Fields and Line Integrals in 3D
Quiz X: on Stoke's Theorem, Conservative Vector Fields and Line Integrals in 3D due Mon 5/13: Exerrcises 7,9,26,29 from handout.
Mon 5/13: Review of Metric Spaces and Compactness
Wed 5/15: Review for final which will cover all material in the course.
In the final you will be given sets and you will prove or disprove statements about whether they are open, closed, compact, connected, bounded and what their diameter is using theorems and definitions that have been taught in the course. You will also be given given functions and asked questions about the images and preimages of the functions, whether the function has an inverse, has a local inverse and whether it has a derivative at a point. For all problems you will be asked to prove or disprove statements. Sample problems will be reviewed in class on Wednesday and additional sample problems will distributed by email. When you receive the email, reply that you have received it.
Final: Monday May 20 6:15-8:15 pm in Gillet 227.