Harvard, Jan 13, 2004Part IIMATH 21B REVIEWContent PreliminariesDiscrete dynamical systems Differential equations Fourier analysis Partial differential equationsSlides to review the conceptsBlackboardproblemsBlackboard problems slower pacedMethodology of this lecture80 slides7 problems•Diagonalization•Complex Numbers•Linear spaces, linear transformations•Differential operatorsI) PreliminariesDiagonalization•A is a symmetric matrix•All eigenvalues of A are differentDiagonalization of a matrix A is possible if: Orthonormal eigenbasisPrototype of nondiagonalizable matrix :Jordan Normal FormComplex NumbersFundamental Theoremof algebraA polynomial of degree n has exactly n roots. If n is odd, there is at least one real root.If a+ib is a root, then a-ib is a root too.Example: Roots of OneExample: permutation matrix:Linear SpacesIn a linear space, we can add and scale. It contains a zero element. To check whether a subset X of one of these spaces is a linear space, we check:x+y is in Xrx is in X0 is in XLinear MapsTo check wether a map between linear spacesis linear, we have to check:T(x+y) =T(x)+T(y), T(rx)=r T(x), T(0)=0Differential operatorsDf=f’p(x) polynomial T=p(D) differential operator Fundamental theorem of calculus: T f = g differential equationInverseExample: T(f) = D f + 10 Df + 21 = (D+3) (D+7)2f’’+10 f’+21 =0 { T(f) =0 } the kernel, is a linear space spanned by e and e{ T(f) =g } not a linear space if g is not zero.f’’+2f’+10 =g -3t -7tThe solution set is formed by adding a particular solution to the kernel. See later.Quizz coming up!2)3)5)7)11)2)3)5)7)11)2114Shout: Win laser pointerStructure of the quizz:TrueI used to have Swiss Army knife prizes. Will soon also laser pointers be banned?Again: here are the rules for the upcoming multiple choice challenge: 10 questions. Encode the answer: 2)3)5)7)11)2)3)5)7)11)2114then shout: TrueIf your choice is:Linear space or not?Linear map or not?42The answer to the Ultimate Question of Life, the Universe and Everything, given by the supercomputer 'Deep Thought' to a group of mice, is "forty-two". "Forty-two!" yelled Loonquawl. "Is that all you've got to show for seven and a half million years' work?" "I checked it very thoroughly,"said the computer, "and that quite definitely is the answer. I think the problem, to be quite honest with you, is that you've never actually known what the question is."•Solving initial value problems•Analyse phase space•Find out about stabilityII) Discrete Dynamical SystemsThe dynamicsEigenvalues The eigenvalues of A determine the stabilty of the origin. Asymptotic stabilityAll eigenvalues have absolute value <1Initial Value Problem•Diagonalize A•Write x as sum of eigenvector•Write down solutionExample Blackboardx(t+1) = x(t) + y(t)y(t+1) = x(t) - y(t)x y224044The Problem: ......•Solving the system•Analyze the phase space•Determine stabilityIII) Differential EquationsDifferential equationsx=f(x,y)y=g(x,y)..We covered a lot of material!But we like it extreme!Linear Differentialequations in the plane..y=cx + dyx=ax + byThe eigenvalues of A determine the dynamical behavior.One dimensional caseThe mother of all linear differential equationsReal, nonzeroeigenvalues..y=cx + dyx=ax + byComplex eigenvaluesSome zeroeigenvalueDo the eigenvalues determine the rotation direction?Mathematics and SexClio Cresswell, 2003Romeo and Juliet..y=-b xx= a y x : love of Romeo to Juliety: love of Juliet towards RomeoRomeo warms up when given more love,Juliet wants to run away when being desired. Problem: analyze, what happens with this love relationship!Romeos loveJuliets loveStability examplesSolved on BlackboardAsymptotically stable?Do not mix up results from discrete dynamical systems with the ones from differential equations! Caution!•Equilibrium points•Nullclines•Nature of equilibrium points•Understanding phase spaceIII) Nonlinear differential equationNonlinear SystemsNull-clinesEqui- librium-pointsx=f(x,y)y=g(x,y)..JacobeanmatrixExampleExample Blackboard1-1•Solving initial value problem p(D)f=g•Homogenous problem p(D)f = 0•Finding special solution to inhomogenous problem.IV) Higher Order Differential EquationsNonhomogeneousdifferential equationsp(D) f(t) = g(t)Three cases for second order differential equations:1) Different real roots of p2) Two identical real roots of p3) Two complex roots of pThe method overview: f’’+5 f = 2 sin(3 x)HomogenousInhomogeneousf’’+5 f=0f(x)= c sin(3 x)f’’+5 f = c+5c=2so c=-1/2f(x)=-sin(3x)/2 +5 =0 = i -if(x)=a sin( x) + b cos( x) f(0)=1, f’(0)=22f’’+5 f = 2 sin(3 x)Problem:Initial conditionsf(0)=1 implies b=1f’(0)=1 implies a=-3/(2 )Solution:f(x)=-3/(2 ) sin( x) + cos( x)-sin(3x)/2Four examplesExamples Blackboardf’-3f=exp(t)f’’+9f=exp(t)f’’+6f’+8f=tf’’-6f’+9f=exp(t)•Fourier Series•Symmetry•ParcevalV) Fourier analysisFourier seriesFourier coefficientsSound synthesis ISound synthesis IISound synthesis IIIFourier approximationEven and Oddcos-seriessin-seriesRemember:Parseval IdentitySome safety tips:ExamAYouWhen doing Fourierintegrals:Seperate into even and odd parts!terms like sin(89 x) are already partof the Fourier decomposition. Dont spend too much time withsimplifying the result. Any correct resultwithout integrals is good enough.A music piece is just a function f(t).We can manipulate thisfunction: i.e. T(f)(t)=f(-t) impliyExampleExample Blackboard•Heat equation•Wave equationVI) Partial DifferentialequationsHeat Evolution0Wave Evolution0ExampleExample Blackboard•Exam is on January 20•Review also midterm exams and homework•Do the practice exams Pro
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