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### Jan24-12

Course: MATH 260, Spring 2012
School: UPenn
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260, Math Spring 2012 Jerry L. Kazdan Class Outline: Jan. 24, 2012 1. Example: quadratic polynomials p(x) with p(1) = 0. 2. Polynomial Interpolation. Find a quadratic polynomial with p(1) = 1, p(2) = 1, p(4) = 3. p2 (x) = x, p3 (x) := x2 . Methods: Naive basis: p1 (x) := 1, Seek p(x) = A1 p1 (x) + A2 p2 (x) + A3 p3 (x) = A1 + A2 x + A3 x2 Lagranges basis: p1 (x) := (x 2)(x 4) , (1 2)(1 4) p2 (x) :=...

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260, Math Spring 2012 Jerry L. Kazdan Class Outline: Jan. 24, 2012 1. Example: quadratic polynomials p(x) with p(1) = 0. 2. Polynomial Interpolation. Find a quadratic polynomial with p(1) = 1, p(2) = 1, p(4) = 3. p2 (x) = x, p3 (x) := x2 . Methods: Naive basis: p1 (x) := 1, Seek p(x) = A1 p1 (x) + A2 p2 (x) + A3 p3 (x) = A1 + A2 x + A3 x2 Lagranges basis: p1 (x) := (x 2)(x 4) , (1 2)(1 4) p2 (x) := (x 1)(x 4) , (2 1)(2 4) p3 (x) := (x 1)(x 2) . (4 1)(4 2) Note p1 (1) = 1 while p1 (2) = p1 (4) = 0, etc. As above, seek p(x) = A1 p1 (x) + A2 p2 (x) + A3 p3 (x). Since p1 (1) = 1 while p2 (1 = p3 (1) = 0, by setting x = 1 we immediately nd that A1 = 1. What are A2 and A3 ? Newtons basis: p1 (x) := 1, p2 (x) = (x 1), p2 (x) := (x 1)(x 2) and seek p(x) = A1 p1 (x) + A2 p2 (x) + A3 p3 (x). Letting x = 1 we nd A1 . Then letting x = 2 we nd A2 etc. Remark To evaluate the interpolating polynomial at intermediate points, Newtons basis uses fewer multiplications. It is also easy to modify it if you want to add an extra interpolating point. If the data represents the values of some unknown function f (x) at the specied points, one often would like understand to the error Error(x) := |f (x) p(x)| . Newtons basis is useful for this too. 3. Least Squares What if you have lots of data points (x1 , y1 ), (x2 , y2 ), . . . , (xk , yk ) 1 and want to nd the straight line y = a + bx that best ts the data? In this case trying interpolation you have k linear equations in the 2 unknowns a and b : a + bx1 =y1 a + bx2 =y2 a + bxk =yk Although it is highly unlikely you will nd an exact solution, what is the best possible approximation? Idea: Pick a and b to minimize the error E (a, b) := (a + bx1 y1 )2 + (a + bx1 y2 )2 + + (a + bx1 yk )2 4. Lu := u + u = 0. a) cos x and sin x for x R are linearly independent. Three methods (all useful): i). naive, ii). using the derivative, integral. b) Show that the dimension of the nullspace of L is at least 2. c) Show that the dimension of the nullspace of L is at most 2. Moral: The dimension of the nullspace is exactly 2. 5. u + u = 2 x . There are two tasks: Find the general solution of the homogeneous equation. Find a particular solution of the inhomogeneous equation. 6. Metrics, Norms, and Inner Products. [Last revised: January 24, 2012] 2 iii). Using the
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UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Jan. 24, 20121. Least Squares What if you have lots of data points(x1 , y1 ), (x2 , y2 ), . . . , (xk , yk )and want to nd the straight line y = a + bx that best ts the data? In this case tryinginte
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Jan. 31, 20121. Metrics, Norms, and Inner Products.Seehttp:/www.math.upenn.edu/kazdan/260S12/notes/math21/math21-2012-2up.pdf(Math 21 Lecture Notes , Chapter 3, p. 101)http:/www.math.upenn.edu/kazd
UPenn - MATH - 260
Many Coupled OscillatorsA V IBRATING S TRINGSay we have n particles with the same mass m equally spaced on a string havingtension . Let yk denote the vertical displacement if the k th mass. Assume the ends of thestring are xed; this is the same as hav
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: March 1, 2012Functions of Several VariablesReading: Marsden and Tromba, Vector Calculus, Chapter 2, Chapter 3.13.4 andSections 8.1-8.2 in the noteshttp:/www.math.upenn.edu/ kazdan/260S12/notes/math2
UPenn - MATH - 260
iPrefaceIntermediate CalculusandLinear AlgebraJerry L. KazdanHarvard UniversityLecture Notes, 19641965These notes will contain most of the material covered in class, and be distributed beforeeach lecture (hopefully). Since the course is an experi
UPenn - MATH - 260
Intermediate CalculusandLinear AlgebraJerry L. KazdanHarvard UniversityLecture Notes, 19641965iPrefaceThese notes will contain most of the material covered in class, and be distributed beforeeach lecture (hopefully). Since the course is an experi
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanMatrices as MapsWe now discuss viewing systems of equations as maps. Think of this an an introduction tocomputer graphics. Well use these ideas throughout Math 260.The standard technique goes back to Descartes intr
UPenn - MATH - 260
Linear ODEsSecond order linear equationsMany traditional problems involving ordinary equations arise as second order linear equationsau + bu + cu = f ,more briey as Lu = f .The problem is, given f , nd u ; often we will want to nd u that satises some
UPenn - MATH - 260
Math 425, Spring 2011Jerry L. KazdanClassical Examples of PDEsLaplace equation:32uu := 2 = 0j=1 x j(some write u = u = 2u)u = f (x)Poisson equation:Helmholtz (or eigenvalue) equation:u=tTransport equation:Heat (or diffusion) equation:Schr
UPenn - MATH - 260
Math 210Jerry L. KazdanQuadratic PolynomialsPolynomials in One Variable.After studying linear functions y = ax + b , the next step is to study quadratic polynomials, y = ax2 + bx + c , whose graphs are parabolas. Initially one studies the simplerspec
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanRepresenting Symmetries by MatricesIf order to understand and work with the symmetries of an object (the symmetries of asquare is a simple example), one would like a way to compute, not just wave your hands.For an
UPenn - MATH - 260
A Tridiagonal MatrixWe investigate the simple n n real tridiagonal matrix: 0 1 0 0 . 0 0 0 0 . 0 0 1 0 1 0 . . . 0 0 0 . . . 0 0 0 1 0 1 . . . 0 0 0 . . . 0 0 . . . . = I + T , . . = I + . . . . . . . . . . M =. . . . . . . . 0 0 0 . . . 0 1 0 0 0 0 . .
UPenn - MATH - 260
Math 210Jerry L. KazdanVectors and an Application to Least SquaresThis brief review of vectors assumes you have seen the basic properties of vectorspreviously.We can write a point in Rn as X = (x1 , . . . , xn ) . This point is often called a vector.
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. Kazdanux + 3uy = 0This example is similar to Problem Set 7 # 3a).Example: Find a function u(x, y ) that satises ux + 3uy = 0 with u(0, y ) = 1 + e2y .Solution:The dierential equation can be written u V = 0 where V = (1,
UPenn - MATH - 260
Math 260Feb. 9, 2012Exam 1Jerry L. Kazdan12:00 1:20Directions This exam has 10 questions (10 points each). Closed book, no calculators orcomputers but you may use one 3 5 card with notes on both sides. Neatness counts.1. Which of the following sets
UPenn - MATH - 260
Math 260Feb. 9, 2012Exam 1Jerry L. Kazdan12:00 1:20Directions This exam has 10 questions (10 points each). Closed book, no calculators orcomputers but you may use one 3 5 card with notes on both sides. Neatness counts.1. Which of the following sets
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanHomework Set 0 [Due: Never]Comples Power SeriesIn our treatment of both dierential equations and Fourier series, it will be essential to usecomples numbers and complex power series. They enormously simplify the sto
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 1Due: In class Thursday, Jan. 19. Late papers will be accepted until 1:00 PM Friday.These problems are intended to be straightforward with not much computation.1. At noon the minute and hour hands of a
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 2Due: In class Thursday, Jan. 26. Late papers will be accepted until 1:00 PM Friday.1. Which of the following sets of vectors are bases for R2 ?a). cfw_(0, 1), (1, 1)d). cfw_(1, 1), (1, 1)b). cfw_(1,
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 3Due: In class Thursday, Feb. 2. Late papers will be accepted until 1:00 PM Friday.1. Say you have k linear algebraic equations in n variables; in matrix form we writeAX = Y . Give a proof or counterexa
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 4Due: Never[Exam 1 is on Thursday, Feb.9, 12:00-1:20]Unless otherwise stated use the standard Euclidean norm.1. In R2 , dene the new norm of a vector V = (x, y ) by V := 2|x| + |y | . Show thissatises
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 5Due: In class Thursday, Feb. 16. Late papers will be accepted until 1:00 PM Friday.Unless otherwise stated use the standard Euclidean norm.1 for x &lt; 0,. Find its Fourier series (either using trig1 fo
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 6Due: In class Thursday, Feb. 23. Late papers will be accepted until 1:00 PM Friday.Unless otherwise stated use the standard Euclidean norm.1. Let u(x, t) be the temperature at time t at a point x on a
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 7Due: In class Thursday, March 1. Late papers will be accepted until 1:00 PM Friday.Unless otherwise stated use the standard Euclidean norm.1. Say you have a matrix A(t) = (aij (t) whose elements aij (t
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanProblem Set 8Due: NeverUnless otherwise stated use the standard Euclidean norm.1. Find a 3 3 symmetric matrix A with the property thatX, AX = x2 + 4x1 x2 x1 x3 + 2x2 x3 + 5x213for all X = (x1 , x2 , x3 ) R3 .2
UPenn - MATH - 240
Problem: Score:12345678 Total:9101112_ Please do not write above this line UNIVERSITY OF PENNSYLVANIA DEPARTMENT OF MATHEMATICS MATH 240 FINAL EXAM Monday December 21, 2009 Your name (printed) _ Signature_ Professor (circle one): Patrick Clar
UPenn - MATH - 240
1Math 240Circle one:FINAL EXAMDecember 17, 2010Professor ShatzProfessor WylieProfessor ZillerName:Penn Id#:Signature:TA:Recitation Day and Time:Please show your work clearly. A correct answer with no work is worth 0 points. Circle your answer
UPenn - MATH - 240
NAME: PROFESSOR: (A) Donagi ; (B) Ghrist ; (C) Krieger240 SPRING 2009: CalculusFINAL EXAM INSTRUCTIONS:1. WRITE YOUR NAME at the top and indicate your professors name. 2. As you solve problems on the exam, FILL IN COMPLETELY the letter(s) of your solut
UPenn - MATH - 240
1Math 240Circle one:FINAL EXAMMay 4, 2010Professor ZillerProfessor ZywinaName:Penn Id#:Signature:TA:Recitation Day and Time:You need to show your work, even for multiple choice problems. A correct answer with no workwill get 0 points. The onl
UPenn - MATH - 240
Math 240 Final Exam, spring 2011Name (printed):TA:Recitation Time:This examination consists of ten (10 problems). Please turn o all electronicdevices. You may use both sides of a 8.5 11 sheet of paper for notes whileyou take this exam. No calculator
UPenn - MATH - 240
Makeup Final Exam, Math 240: Calculus IIISeptember, 2011No books, papers or may be used, other than a hand-writtennote sheet at most 8.5 11 in size. All electronic devicesmust be switched o during the exam.This examination consists of nine (9) long a
UPenn - MATH - 425
Math 425March 3, 2011Exam 1Jerry L. Kazdan12:00 1:20Directions This exam has three parts, Part A, short answer, has 1 problem (12 points). Part Bhas 5 shorter problems (7 points each, so 35 points). Part C has 3 traditional problems (15 pointseach
UPenn - MATH - 425
Math 425March 3, 2011Exam 1Jerry L. Kazdan12:00 1:20Directions This exam has three parts, Part A, short answer, has 1 problem (12 points). Part Bhas 5 shorter problems (7 points each, so 35 points). Part C has 3 traditional problems (15 pointseach
UPenn - MATH - 425
SignaturePrinted NameMy signature above certies that I have complied with the University of PennsylvaniasCode of Academic Integrity in completing this examination.Exam 2Math 425April 26, 2011Jerry L. Kazdan12:00 1:20Directions This exam has three
UPenn - MATH - 425
SignaturePrinted NameMy signature above certies that I have complied with the University of PennsylvaniasCode of Academic Integrity in completing this examination.Exam 2Math 425April 26, 2011Jerry L. Kazdan12:00 1:20Directions This exam has three
UPenn - MATH - 425
AMSIJan. 14 Feb. 8, 2008Partial Differential EquationsJerry L. KazdanCopyright c 2008 by Jerry L. Kazdan[Last revised: March 28, 2011]ivCONTENTS7. Dirichlets principle and existence of a solutionChapter 6. The RestContentsChapter 1. Introductio
UPenn - MATH - 425
Math 425Notes and exercises on Black-ScholesDr. DeTurckApril 2010On Thursday we talked in class about how to derive the Black-Scholes dierentialequation, which is used in mathematical nance to assign a value to a nancial derivative. The latter is usu
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. Kazdanwhere the coefcient matrix B := (bk ) isPDE: Linear Change of Variablenbk =Let x := (x1 , x2 , . . . , xn ) be a point inential operatorai, j ski s j .i, j=1Rnand consider the second order linear partial diff
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanPDE: Linear Change of VariableLet x := (x1, x2, . . . , xn) be a point in Rn and considerthe second order linear partial differential operatorn2 uLu := ai j,xix ji, j=1(1)where the coefcient matrix A := (ai
UPenn - MATH - 425
Math 425 Midterm 1Dr. DeTurck March 2, 2010There are four problems on this test. You may use your book and your notes during this exam. Do as much of it as you can during the class period, and turn your work in at the end. But take the sheet with the pr
UPenn - MATH - 425
Math 425 Midterm 1Dr. DeTurck February 8, 20071. Solve ux + yuy + u = 0, u(0, y ) = y . In what domain in the plane is your solution valid? 2. Let u(x, t) be the temperature in a rod of length L that satises the partial dierential equation: ut = kuxx fo
UPenn - MATH - 425
Fourier Series of f (x) = xTo write:eikx.x = ck2kThe Fourier coefcients are112i ck = xeikx dx = x[cos kx i sin kx] dx = x sin kx dx2 2 2 0Butx cos kx 1 cos kx sin kx dx =+cos kx dx == (1)k .kk0kk00Thus2i(1)kck = (1)k =
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 0 [Rust Remover]D UE : Never.1. Let u(t ) be the amount of a radioactive element at time t and say initially, u(0) = A . The rateof decay is proportional to the amount present, sodu= cu(t ),dtwhere
UPenn - MATH - 425
Math 425/525, Spring 2011Jerry L. KazdanProblem Set 1D UE : In class Thursday, Jan. 27. Late papers will be accepted until 1:00 PM Friday.1. Find Greens function g(x, s) to get a formula u(x) =of u (x) = f (x) .Rx0g(x, s) f (s) ds for a particular
UPenn - MATH - 425
Math 425/525, Spring 2011Jerry L. KazdanProblem Set 2D UE : In class Thursday, Feb. 3 Late papers will be accepted until 1:00 PM Friday.1. Find the solution U (t ) := (u1 (t ), u2 (t ) ofu1 =u1u2 =u1 u2with the initial conditions U (0) = (u1 (0), u
UPenn - MATH - 425
Math 425Assignment 3Dr. DeTurckDue Tuesday, February 9, 2010Reading: Textbook, Chapter 1, skim through Chapter 2.1. Take a moment to review the divergence theorem from vector calculus, then work thefollowing problem: Suppose V (x, y, z ) is a vector
UPenn - MATH - 425
Math 425/525, Spring 2011Jerry L. KazdanProblem Set 3D UE : In class Thursday, Feb. 10 Late papers will be accepted until 1:00 PM Friday.1. Solve ux + uy + u = ex+2y with u(x, 0) = 0.2. Find the general solution of uxy = x2 y for the function u(x, y)
UPenn - MATH - 425
Math 425/525, Spring 2011Jerry L. KazdanProblem Set 4D UE : In class Thursday, Feb. 17 Late papers will be accepted until 1:00 PM Friday.1. Solve the wave equation utt = c2 uxx for the semi-innite string x 0 with the initial andboundary conditionsu(
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 5D UE : Thurs. Feb. 24 Late papers will be accepted until 1:00 PM Friday.1. In R4 the vectorsU1 := (1, 1, 1, 1),U2 := (1, 1, 1, 1),U3 := (2, 2, 2, 2),U4 := (1, 1, 1, 1)are orthogonal, as you can eas
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 6D UE : Thursday March 17 [Late papers will be accepted until 1:00 PM Friday].1. This problem concerns orthogonal projections into a subspace of a larger space.a) Let U = (1, 1, 0, 1) and V = (1, 2, 1,
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 7D UE : Thursday March 24 [Late papers will be accepted until 1:00 PM Friday].1. Suppose u is a twice differentiable function on R which satises the ordinary differentialequationu + b(x)u c(x)u = 0,wh
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 8D UE : Thursday March 31 [Late papers will be accepted until 1:00 PM Friday].1. a) Let A be an n n invertible real symmetric matrix, b Rn and c R . For x Rn considerthe quadratic polynomialQ(x) = x, A
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 9D UE : Thursday April 7 [Late papers will be accepted until 1:00 PM Friday].1. a) In a bounded region Rn , let u(x, t ) satisfy the modied heat equationut = u + cu,where c is a constant,(1)as well a
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanProblem Set 10D UE : Thursday April 14 [Late papers will be accepted until 1:00 PM Friday].1. This problem is to help with a computation in class today (Thursday) nding a formula for aparticular solution of the inh
UPenn - MATH - 425
INTERMEDIATE CALCULUSANDLINEAR ALGEBRAPart IJ. KAZDANHarvard UniversityLecture NotesiiPrefaceThese notes will contain most of the material covered in class, and be distributed beforeeach lecture (hopefully). Since the course is an experimental o
UPenn - MATH - 425
UPenn - MATH - 425
UPenn - MATH - 425
First-order ordinary differential equations Before we get involved trying to understand partial differential equations (PDEs), we'll review the highlights of the theory of ordinary differential equations (ODEs). We'll do this in such a way that we can beg
UPenn - MATH - 425
Linear ODEsSecond order linear equationsMany traditional problems involving ordinary equations arise as second order linear equationsau + bu + cu = f ,more briey asLu = f .The problem is, given f , nd u ; often we will want to nd u that satises some
UPenn - MATH - 425
Math 425 Midterm 1Dr. DeTurck February 8, 20071. Solve ux + yuy + u = 0, u(0, y ) = y . In what domain in the plane is your solution valid? As usual, we construct the graph of the solution by propagating the initial data o the line in the xy -plane wher
UPenn - MATH - 425
Math 425/525, Spring 2011Jerry L. KazdanPeriodic Solutions of ODEsIn class we discussed some aspects of periodic solutions of ordinary differential equations. Fromthe questions I received, my presentation was not so clear. Here Ill give a detailed for