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Course: MATH 260, Spring 2012
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260, Math Spring 2012 Jerry L. Kazdan Linear Maps from R2 to R3 As an exercise, which I hope you will (soon) realize is entirely routine, we will show that a linear map F (X ) = Y from R2 to R3 must just be three linear high school equations in two variables: a11 x1 + a12 x2 = y1 a21 x1 + a22 x2 = y2 (1) a31 x1 + a32 x2 = y3 Linearity means for any vectors U and V in R2 and any scalars c F (U + V ) = F (U ) +...

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260, Math Spring 2012 Jerry L. Kazdan Linear Maps from R2 to R3 As an exercise, which I hope you will (soon) realize is entirely routine, we will show that a linear map F (X ) = Y from R2 to R3 must just be three linear high school equations in two variables: a11 x1 + a12 x2 = y1 a21 x1 + a22 x2 = y2 (1) a31 x1 + a32 x2 = y3 Linearity means for any vectors U and V in R2 and any scalars c F (U + V ) = F (U ) + F (V ) and F (cU ) = cF (U ). Idea : write X := (x2 , x2 ) R2 as X = x1 (1, 0) + x2 (0, 1) = x1 e1 + x2 e2 , where e1 := (1, 0), e2 := (0, 1) (physicists often write e1 i as and e2 as j but using this notation in higher dimensions one quickly runs out of letters). Then, by the two linearity properties Y = F (X ) =F (x1 e1 + x2 e2 ) =F (x1 e1 ) + F (x2 e2 ) =x1 F (e1 ) + x2 F (e2 ). But F (e1 ) and F (e2 ) are just specic vectors in R3 so this last equation is exactly the desired (1) with a12 a11 a22 . a21 and F (e2 ) = F (e1 ) = a32 a31 Collecting the ingredients we have found that x1 a11 + x2 a12 a12 a11 y1 y2 = Y = F (x) = x1 a21 + x2 a22 = x1 a21 + x2 a22 a31 a32 x1 a31 + x2 a32 y3 as claimed in(1). [Last revised: January 13, 2012] 1
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UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Feb. 2, 20121. More on Metrics, Norms, and Inner Products.We will continue to discuss orthogonal projections with applications to Fourier Seriesand the Method of Least Squares. For some details see:
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Feb. 7, 20121. In honor of the Exam on Thursday, the rst part of todays class will be a review ofthe course so far.2. More on Inner Products.We will continue to discuss orthogonal projections with a
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Feb. 14, 20121. In honor of the Exam last Thursday, the rst part of todays class will discuss thisexam, whose solutions are now posted athttp:/www.math.upenn.edu/%7Ekazdan/260S12/260S12Ex1s-solns.pdf
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Feb. 16, 20121. Curves in Space We will consider the motion of a particle in R3 , say its position isgiven by the vector function X (t).a) Concepts: The derivative, tangent vector, velocity vector, s
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Feb. 21, 2012Scalar Functions of Several VariablesReading: Marsden and Tromba, Vector Calculus, Chapter 2.1. Directional Derivative special case: partial derivatives.Example: Linear polynomial f (X
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Feb. 2, 2012Scalar Functions of Several VariablesReading: Marsden and Tromba, Vector Calculus, Chapter 2, Chapter 3.13.4 andSections 8.1-8.2 inhttp:/www.math.upenn.edu/ kazdan/260S12/notes/math21/ma
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Feb. 28, 2012Scalar Functions of Several VariablesReading: Marsden and Tromba, Vector Calculus, Chapter 2, Chapter 3.13.4 andSections 8.1-8.2 inhttp:/www.math.upenn.edu/ kazdan/260S12/notes/math21/m
UPenn - MATH - 260
Fourier Series An ExampleFormulas: Let f(x) be periodic with period 2 . We want to writeA0+2f (x) =(Ak cos kx + Bk sin kx) .1The Fourier coecients are given by the formulas11f (x) cos kx dxBk =f (x) sin kx dx.Ak = Moreover one has the an
UPenn - MATH - 260
Fourier Series of f (x) = xGiven a real periodic function f (x) , &lt; x &lt; , one can nd its Fourier series in two(equivalent) ways: using trigonometric functions:cos kxsin kxa0f (x) = + ak + bk 2 k=1or using the complex exponentialf (x) =eikxck .
UPenn - MATH - 260
Math 425, Spring 2011Jerry L. KazdanInner Product SummaryVN = span cfw_1, cos x, cos 2x, . . . , cos Nx, sin x, . . . , sin Nx.An orthonormal basis is:This is a summary of some items from class on Tues, Feb. 15, 2011.S ETTING : Linear spaces X , Y w
UPenn - MATH - 260
Math 425, Spring 2011Jerry L. KazdanInner Product SummaryThis is a summary of some items from class on Tues, Feb. 15, 2011.S ETTING : Linear spaces X , Y with inner products ,Example: X = R4 and Y = R7 .Vectors x, z X are orthogonal if x, zXXand
UPenn - MATH - 260
Math 425, Spring 2011Jerry L. KazdanInner Product SummaryThis is a summary of some items from class on Tues, Feb. 15,2011.S ETTING : Linear spaces X , Y with inner products , X and, Y.Example: X = R4 and Y = R7 .Vectors x, z X are orthogonal if x,
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Jan. 19, 20121. Definitions: Homogeneous equation, inhomogeneous equation.Basic Lemma: If you have n linear algebraic equations in k unknowns, and if n &gt; k(so more equations than unknowns), then the
UPenn - MATH - 260
Math 260, Spring 2012Jerry L. KazdanClass Outline: Jan. 24, 20121. Example: quadratic polynomials p(x) with p(1) = 0.2. Polynomial Interpolation. Find a quadratic polynomial withp(1) = 1,p(2) = 1,p(4) = 3.p2 (x) = x,p3 (x) := x2 .Methods:Naive
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)