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Course: MATH 425, Spring 2011
School: UPenn
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425 Math Midterm 1 Dr. DeTurck February 8, 2007 1. 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 for (x, t) (0, L) (0, ), where k is a positive constant, together with the initial condition u(x, 0) = (x) for x [0, L], where satises (0) = (L) = 0 and (x)...

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425 Math Midterm 1 Dr. DeTurck February 8, 2007 1. 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 for (x, t) (0, L) (0, ), where k is a positive constant, together with the initial condition u(x, 0) = (x) for x [0, L], where satises (0) = (L) = 0 and (x) > 0 for x (0, L). (a) If u also satises the Neumann boundary conditions ux (0, t) = 0, ux (L, t) = 0, show that the average temperature in the rod at time t, which is given by A(t) = is a constant (independent of t). (b) On the other hand, if u satises the Dirichlet boundary conditions u(0, t) = 0, u(L, t) = 0, 1 L L u(x, t) dx 0 show it that must be the case the u(x, t) 0 for all (x, t) satisfying 0 < x < L and t > 0. (c) Still under the assumption that u satises the Dirichlet boundary conditions, show that A(t) is a non-increasing function of t. (Hint for (a) and (c): Use an argument similar to an energy argument). 3. (a) Solve the wave equation with friction: uxx = utt + 2ut for 0 < x < and t > 0 with the initial conditions u(x, 0) = sin x, ut (x, 0) = 0, and the boundary conditions u(0) = u( ) = 0. (Hint: Look for separated solutions) (b) If E (t) = show that t 1 2 u2 + u2 dx, t x 0 lim E (t) = 0. (Hint: To do this, you can calculate E (t) explicitly). 4. Find as general a solution u(x, y, z ) as you can to the third-order equation uxyz = 0
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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
UPenn - MATH - 425
Math 425Practice Midterm 1Dr. DeTurckFebruary 20101. Suppose f is a function of one variable that has a continuous second derivative. Show thatfor any constants a and b, the functionu(x, y ) = f (ax + by )is a solution of the PDEuxx uyy u2 = 0.xy
UPenn - MATH - 425
Math 425Hints and Solutions to Practice Midterm 1Dr. DeTurckFebruary 20101. Suppose f is a function of one variable that has a continuous second derivative. Show thatfor any constants a and b, the functionu(x, y ) = f (ax + by )is a solution of the
UPenn - MATH - 425
Math 425, Spring 2011Jerry L. KazdanThe Wave Equation in R2 and R3To Solveutt = c2u,x Rnwithu(x, 0) = (x),ut (x, 0) = (x)In R3 [Poisson: Kirchoffs formula]1u(x0, t0) =4c2t01(x) dS +t0 4c2t0SZZZZ(x) dS ,Swhere S is the sphere centered
UPenn - MATH - 508
Exam 1Math 508October 12, 2006Jerry L. Kazdan12:00 1:20Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems(6
UPenn - MATH - 508
Exam 1Math 508October 12, 2006Jerry L. Kazdan12:00 1:20Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems(6
UPenn - MATH - 508
SignaturePrinted NameExam 2Math 508December 8, 2006Jerry L. Kazdan12:00 1:20Directions This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points),Part B has 5 traditional problems (15 points each, so 75 points).Closed bo
UPenn - MATH - 508
Exam 2Math 508December 8, 2006Jerry L. Kazdan12:00 1:20Directions This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points),Part B has 5 traditional problems (15 points each, so 75 points).Closed book, no calculators but
UPenn - MATH - 508
Exam 1Math 508October 16, 2008Jerry L. Kazdan10:30 11:50Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems(
UPenn - MATH - 508
Exam 2Math 508December 4, 2008Jerry L. Kazdan10:30 11:50Directions This exam has two parts, Part A has 10 True-False problems (30 points, 3 pointseach). Part B has 5 traditional problems (70 points, 14 points each).Closed book, no calculators or co
UPenn - MATH - 508
Exam 1Math 508October 14, 2010Jerry L. Kazdan9:00 10:20Directions This exam has three parts, Part A asks for 4 examples (20 points, 5 points each).Part B has 4 shorter problems (36 points, 9 points each. Part C has 3 traditional problems (45points,
UPenn - MATH - 508
Exam 2Math 508December 9, 2010Jerry L. Kazdan9:00 10:20Directions This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points).Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15p
UPenn - MATH - 508
Math 508December 9, 2010Exam 2Jerry L. Kazdan9:00 10:20Directions This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points).Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15p
UPenn - MATH - 508
Advanced Analysis: OutlineMath 508, Fall 2010Jerry L. KazdanThis outline of the course is to help you step back and get a larger view of what we havedone. Since this is only an outline, I will often not explicitly state the precise assumptionsneeded
UPenn - MATH - 508
Analysis ProblemsPenn MathJerry L. KazdanIn the following, when we say a function is smooth, we mean that all of its derivatives existand are continuous.These problems have been crudely sorted by topic but this should not be taken seriouslysince man
UPenn - MATH - 508
Some Classical InequalitiesFor all of these inequalities there are many methods. We give a sampling. 1.ARITHMETIC - GEOMETRIC MEAN and decide when equality holds. INEQUALITYIf cfw_ b j &gt; 0, prove the following (1)(b1 b2 bn )1/n b1 + b 2 + + b n . n
UPenn - MATH - 508
Basic DenitionsLet S Rn . and p Rn . S is bounded if it is contained in someball in Rn . S is a neighborhood of p if S containssome open ball around P . A point p is a limit point of S if everyneighborhood of p contains a point q S , where q = p .
UPenn - MATH - 508
Calculus ProblemsMath 504 505Jerry L. Kazdan1. Sketch the points (x , y) in the plane R2 that satisfy |y x| 2.2. A certain function f (x) has the property thatf (t ) dt = ex cos x + C . Find both f0and the constant C .cos x3. Compute limx0 cos 2
UPenn - MATH - 508
Math 508Jerry L. KazdanCompleteness of1Let 1 be the vector space of innite sequences of real numbers X = (x1 , x2 , . . .) with nite normX := =1 |x j | . Here we show this space is complete. The proof is a bit fussy.j(n)(n)S TEP 1: F IND A CANDID
UPenn - MATH - 508
Contracting Maps and an ApplicationMath 508, Fall 2010Jerry L. KazdanOne often effective way to show that an equation g(x) = b has a solution is to reduce theproblem to nd a xed point x of a contracting map T , so T x = x . For instance, assumeV is a
UPenn - MATH - 508
ConvolutionLet f (x) and g(x) be continuous real-valued functions for x R and assume that f or g is zerooutside some bounded set (this assumption can be relaxed a bit). Dene the convolution( f g)(x) :=Zf (x y)g(y) dy(1)Since f or g is zero outside
UPenn - MATH - 508
Numbers and Sets - exercises for enthusiasts 1.W. T. G.1. Let A be the sum of the digits of 44444444 , and let B be the sum of the digits of A.What is the sum of the digits of B ?nn2. Let x1 , . . . , xn be real numbers such that i=1 xi = 0 and i=1
UPenn - MATH - 508
Numbers and Sets - exercises for enthusiasts 2.W. T. G.1. Does there exist an uncountable family B of subsets of N such that for every A, B B(distinct) the intersection of A with B is nite?2. Is it possible to write the closed interval [0, 1] as a cou
UPenn - MATH - 508
The language and grammar ofmathematics1gate and multiply and render the sentences unintelligible.To illustrate the sort of clarity and simplicitythat is needed in mathematical discourse, let usconsider the famous mathematical sentence Twoplus two e
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 0: Rust RemoverD UE : These problems will not be collected.You should already have the techniques to do these problems, although they may take somethinking.1. Show that for any positive integer n , the nu
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 1D UE : Thurs. Sept. 16, 2010. Late papers will be accepted until 1:00 PM Friday.1. Let x0 = 1 and dene xk :=increasing.2. Show that 1 +3xk1 + 4, k = 1, 2, . . . . Show that xk &lt; 4 and that the xk are1
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 2D UE : Thurs. Sept. 23, 2010. Late papers will be accepted until 1:00 PM Friday.1. Let F be a eld, such as the reals or the integers mod 7 and x, y F . Here x means theadditive inverse of x .a) If xy = 0
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 3D UE : Thurs. Sept. 30, 2010. Late papers will be accepted until 1:00 PM Friday.1. Find all (complex) roots z = x + iy of z2 = i .2. Let xn &gt; 0 be a sequence of real numbers with the property that they co
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 4D UE : Thurs. Oct 7, 2010. Late papers will be accepted until 1:00 PM Friday.5n + 17.n+23n2 2n + 17. Calculate lim an .b) Let an := 2nn + 21n + 21. a) Calculate limn2. Investigate the convergence
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanBonus Problem for Set 41. Dene two real numbers x and y to be equal if |x y| is an integer. We write x y (mod 1) .Thus we have a topological circle whose circumference is one.Let be an irrational real number, 0 &lt; &lt; 1
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 5D UE : Thurs. Oct. 21, 2010. Late papers will be accepted until 1:00 PM Friday.k1. [Ratio Test] Let ak be a sequence of complex numbers. Ley s := lim sup aa+1 . By comparisonkwith a geometric series, sh
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 6D UE : Thurs. Oct. 28, 2010. Late papers will be accepted until 1:00 PM Friday.1. Give examples of the following:a) An open cover of cfw_x R : 0 &lt; x 1 that has no nite sub-cover.b) A metric space having
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 7D UE : Thurs. Nov. 4, 2010. Late papers will be accepted until 1:00 PM Friday.Note: We say a function is smooth if its derivatives of all orders exist and are continuous.1. Let f : [a, ) R be a smooth fun
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 8D UE : Thurs. Nov. 11, 2010. Late papers will be accepted until 1:00 PM Friday.Note: We say a function is smooth if its derivatives of all orders exist and are continuous.1. a) Let A(t ) and B(t ) be n n
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 9D UE : Thurs. Nov. 18, 2010. Late papers will be accepted until 1:00 PM Friday.Note: We say a function is smooth if its derivatives of all orders exist and are continuous.1. Let f (x) be a smooth function
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 10D UE : Tues. Nov. 30, 2010. Late papers will be accepted until 1:00 PM Wednesday.Note: We say a function is smooth if its derivatives of ball orders exist and are continuous.1. Find an integer N so thst
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 11D UE : NeverNote: We say a function is smooth if its derivatives of ball orders exist and are continuous.1. Partition [a, b] R into sub-intervals a &lt; x1 &lt; x2 &lt; &lt; xn = b . A function h(x) that isconstant
UPenn - MATH - 508
Math 508, Fall 2008Jerry KazdanTwo Inequalities for Integrals of Vector Valued FunctionsTheorem Let F : [a, b] Rn be a continuous vector-valued function. ThenZbF (t ) dt ZbF (t ) dtaawith equality if andR only if there is a continuous scalar val
UPenn - MATH - 508
NUMBERS AND SETS EXAMPLES SHEET 1.W. T. G.1. Let A, B and C be three sets. Give a proof that A \ (B C ) = (A \ B ) (A \ C ) usingthe criterion for equality of sets.2. The symmetric dierence AB of A and B is dened to be (A \ B ) (B \ A). (Thatis, it
UPenn - MATH - 508
M 2003NUMBERS AND SETS EXAMPLES SHEET 2W. T. G.1. Prove by induction that the following two statements are true for every positive integer n.(i) The number 2n+2 + 32n+1 is a multiple of 7.(ii) 13 + 33 + 53 + . . . + (2n 1)3 = n2 (2n2 1) .2. Suppose
UPenn - MATH - 508
NUMBERS AND SETS EXAMPLES SHEET 3.W. T. G.1. Solve (ie., nd all solutions of) the equations(i) 7x 77 (mod 40).(ii) 12y 30 (mod 54).(iii) 3z 2 (mod 17) and 4z 3 (mod 19).2. Without using a calculator, work out the value of 1710,000 (mod 31).3. Again
UPenn - MATH - 508
NUMBERS AND SETS EXAMPLES SHEET 4.W. T. G.1. Is there a eld with exactly four elements? Is there one with six elements?2. Let F be a eld. Prove that (1)(1) = 1 in F. [1 is of course dened to be theadditive inverse of the multiplicative identity. If yo