Linearity, linear operators, and self adjoint eigenvalue problems


 Kelly Snow
 2 years ago
 Views:
Transcription
1 Linerity, liner opertors, nd self djoint eigenvlue problems 1 Elements of liner lgebr The study of liner prtil differentil equtions utilizes, unsurprisingly, mny concepts from liner lgebr nd liner ordinry differentil equtions. Here brief overview of the required concepts is provided. 1.1 Vector spces nd liner combintions A vector spce S is set of elements  numbers, vectors, functions  together with notions of ddition, sclr multipliction, nd their ssocited properties. A liner combintion of elements v 1, v 2,..., v k S is ny expression of the form c 1 v 1 + c 2 v c k v k where the coefficients c j re rel or complex sclrs. A significnt property of vector spces is tht ny liner combintion of elements in S is lso in S. This is esily verified in most cses  for exmple, R n the set of ndimensionl vectors nd C R the set of continuous functions on the rel line re vector spces. 1.2 Liner trnsformtions nd opertors Suppose A is n n mtrix, nd v is ndimensionl vector. The mtrixvector product y Av cn be regrded s mpping tht tkes v s input nd produces the ndimensionl vector y s n output. More precisely this mpping is liner trnsformtion or liner opertor, tht tkes vector v nd trnsforms it into y. Conversely, every liner mpping from R n R n is represented by mtrix vector product. The most bsic fct bout liner trnsformtions nd opertors is the property of linerity. In words, this sys tht trnsformtion of liner combintion is the liner combintion of the liner trnsformtions. This clerly pplies to the cse where the trnsformtion is given by mtrix A, since Ac 1 v 1 + c 2 v 2 c 1 Av 1 + c 2 Av 2. 1 More generlly, if f 1, f 2 re elements of vector spce S, then liner opertor L is mpping from S to some other vector spce frequently lso S so tht 1.3 Eigenvlues nd eigenvectors Recll tht v, λ is n eigenvectoreigenvlue pir of A if Lc 1 f 1 + c 2 f 2 c 1 Lf 1 + c 2 Lf 1. 2 Av λv. Except in rre cses for so clled nonnorml mtrices, there re exctly n linerly independent eigenvectors v 1, v 2,..., v n corresponding to eigenvlues λ 1, λ 2,..., λ n the eigenvlues cn sometimes be the sme for different eigenvectors, however. Why is this useful fct? Note tht if x is ny ndimensionl vector, it cn be written in terms of the bsis given by the eigenvectors, i.e. x c 1 v 1 + c 2 v c n v n. 3 1
2 The c i solve liner system Vc x where V hs columns of eigenvectors nd c is the vector c 1 c 2... T. When A is selfdjoint see below, there is much esier wy to find the coefficients. Note tht ction of liner trnsformtion A on the vector x cn be written simply s Ax Ac 1 v 1 + c 2 v c n v n c 1 Av 1 + c 2 Av c n Av n c 1 λ 1 v 1 + c 2 λ 2 v c n λv n. In other words, eigenvectors decompose liner opertor into liner combintion, which is fct we often exploit. 1.4 Inner products nd the djoint opertor It is frequently helpful to ttch geometric ides to vector spces. One wy of doing this is to specify n inner product, which is mp S S R or S S C. The inner product is bsiclly wy of specifying how to mesure ngles nd lengths. For v 1, v 2 S, we will write n inner product s v 1, v 2. There re mny different wys to define inner products, but they must ll hve the following properties: 1. Symmetry: v, u u, v for every u, v S. If the inner product is complex vlued, this needs to be v, u u, v, where z denotes the complex conjugte. 2. Linerity in the first vrible: for ny vector v nd vectors v 1, v 2,..., v n we hve c 1 v 1 + c 2 v c n v n, v c 1 v 1, v + c 2 v 2, v c n v n, v Positivity: v, v > unless v. Note for complex vector spces, the first two properties imply conjugte linerity in the second vrible: v, c 1 v 1 + c 2 v c n v n c 1 v, v 1 + c 2 v, v c n v, v n. 5 The inner product defines length in the sense tht v, v is thought of s the squre of the mgnitude or norm of v. The inner product lso mesures how prllel two elements of vector spce re. In prticulr, we define v 1, v 2 to be orthogonl if v 1, v 2. Once n inner product is defined, then for ny liner trnsformtion or opertor L, there is nother opertor clled the djoint of L, written L. Wht defines the djoint is tht, for ny two vectors v 1, v 2, Lv 1, v 2 v 1, L v 2. 6 This definition is bit confusing becuse L is not explicitly constructed. You should think of this s if I find n opertor L tht stisfies property 6, it must be the djoint. The dot product for finite dimensionl vectors is the best known exmple of n inner product there re in fct mny wys of defining inner products, even for vectors. As n exmple, consider the liner opertor on ndimensionl rel vectors given by multiplying by mtrix, i.e. Lv Av. By the usul rules of mtrix multipliction Av 1 v 2 v 1 A T v 2, which mens tht the trnspose of A is sme s the djoint of A, with respect to the inner product defined by the dot product. Note tht if one used different inner product, the djoint might be different s well. 2
3 1.5 Selfdjointness nd expnsion in eigenvectors Sometimes n opertor is its own djoint, in which cse its clled selfdjoint. Selfdjoint opertors hve some very nice properties which we will exploit. The most importnt re 1. The eigenvlues re rel. 2. The eigenvectors corresponding to different eigenvlues re orthogonl. Suppose mtrix A is symmetric nd therefore selfdjoint, nd we know its eigenvectors. As in 3, we cn try to write ny vector x s liner combintion of the eigenvectors. Tking the inner product of 3 with ny prticulr eigenvector v k nd using 4, we hve x, v k c 1 v 1 + c 2 v c n v n, v n c 1 v 1, v k + c 2 v 2, v k c 1 v n, v k c k v k, v k 7 since v k is orthogonl to ll eigenvectors except itself. Therefore we hve simple formul for ny coefficient c k : c k x, v k v k, v k. 8 In some cses the eigenvectors re rescled or normlized so tht v k, v k 1, which mens tht 8 simplifies to c k x, v k. 2 Differentil liner opertors We cn think of derivtives s liner opertors which ct on vector spce of functions. Although these spces re infinite dimensionl recll, for instnce, tht tht 1, x, x 2,... re linerly independent, notions such s linerity, eigenvlues, nd djoints still pply. As n exmple, consider the second derivtive of function d 2 f/dx 2. This cn be thought of s mpping fx to the output f x. The bsic linerity property 1 is esily verified since d 2 d 2 f 1 dx 2 c 1 f 1 x + c 2 f 1 x c 1 dx 2 + c d 2 f 2 2 dx 2. There is technicl point to be mde here. We sometimes hve to worry bout wht set of functions ctully constitutes our vector spce. For the exmple bove, nturl vector spce for the domin of the liner opertor is C 2 R, the set of ll twice differentible functions on the rel line. We could hve insted chosen the set of infinitely differentible functions C R. Chnging the spce of functions on which differentil opertor cts my ffect things like eigenvlues nd djointness properties. 2.1 Liner differentil equtions All liner equtions involve liner opertor L. There re two types of liner equtions, homogeneous nd inhomogeneous, which hve the forms Lf, homogeneous, Lf g, inhomogeneous. 3
4 Here f is the solution the function to be found, L is some differentil liner opertor, nd g is nother given function. As rule of thumb, identifying liner eqution is just mtter of mking sure tht ech term in the eqution is liner opertor cting on the unknown function, or term which does not involve the unknown. 2.2 The superposition principle The big dvntge of linerity is the superposition principle, which permits dividendconquer strtegy for solving equtions. If we cn find enough linerly independent solutions, we cn get ll solutions simply by forming liner combintions out of these building blocks. More precisely, Superposition principle for homogeneous equtions: If f 1, f 2,... re solutions of Lf, then so is ny liner combintion: Lc 1 f 1 + c 2 f c 1 Lf 1 + c 2 Lf This bsic ide cn be mended for inhomogeneous equtions. In this cse, one needs to find ny single prticulr solution f p which solves Lf p g, so tht the difference h f f p solves homogeneous eqution Lh. The result is Superposition principle for inhomogeneous equtions: If h 1, h 2,... re solutions of Lh, then ny liner combintion of these plus the prticulr solution solves the inhomogeneous eqution Lf g: Lf p + c 1 h 1 + c 2 h Lf p + c 1 Lh 1 + c 2 Lh g. 2.3 Superposition principle nd boundry conditions One difficulty tht rises from using the superposition principle is tht, while the eqution itself my be stisfied, not every liner combintion will stisfy the boundry or initil conditions unless they re of specil form. We cll ny side condition liner homogeneous if it cn be written in the form Bf, where B is liner opertor. The most bsic types of boundry conditions re of this form: for the Dirichlet condition, B is the restriction opertor which simply mps function to its boundry vlues. For the Neumnn condition, the opertor first tkes norml derivtive before restriction to the boundry. We my extend the superposition principle to hndle boundry conditions s follows. If f 1, f 2,... stisfy liner, homogeneous boundry conditions, then so does n superposition of these c 1 f 1 +c 2 f Therefore in most PDEs, the superposition principle is only useful if t lest some of the side conditions re liner nd homogeneous. 2.4 SturmLiouville opertors As n illustrtion, we will consider clss of differentil opertors clled SturmLiouville opertors tht re firly esy to work with, nd frequently rise in the study of ordinry nd prtil differentil equtions. For specified coefficient functions qx nd px >, these opertors hve the form L d px d + qx. 9 dx dx the nottion mens tht to pply L to function fx, we put f on the right hnd side nd distribute terms so Lf pxf + qxfx The functions p nd q re prescribed, nd in the 4
5 most bsic cses re simply constnt. The vector spce tht L cts on will be C [, b], which mens tht f is infinitely differentible nd f fb i.e. homogeneous Dirichlet boundry conditions. 2.5 Inner products nd self djointness As pointed out erlier, there re mny wys of defining n inner product. Given two functions f, g in C [, b], we define the so clled L2 inner product s f, g fxgxdx. 1 We cn now show tht the SturmLiouville opertor 9 cting on C [, b] is selfdjoint with respect to this inner product. How does one compute the djoint of differentil opertor? The nswer lies in using integrtion by prts or in higher dimensions, Green s formul. For ny two functions f, g in C [, b] we hve Lf, g f, Lg. d px df gx + qxfxgxdx dx dx px df dg df + qxfxgxdx + pxgx b dx dx dx d px dg fx + qxfxgxdx pxfx dg dx dx dx Integrtion by prts ws used twice to move derivtives off of f nd onto g. Therefore compring to 6 L is its own djoint. Note tht the boundry conditions were essentil to mke the boundry terms in the integrtion by prts vnish. As nother exmple, consider the weighted inner product on C [, R] defined by f, g rfrgrdr. 11 Then for the liner opertor Lf r 1 rf, integrtion by prts twice gives Lf, g f, Lg. [ r r 1 d r df dr dr r df dg dr dr dr, d r dg dr dr [ r r 1 d dr fr dr, r dg dr ] grdr int. by prts ] fr dr int. by prts Note in the finl step, weight fctor of r ws needed inside the integrl to obtin the inner product 11. Thus L is self djoint with respect to the weighted inner product. It turns out not to be self djoint, however, with respect to the unweighted one 1. b 5
6 Wht bout nonselfdjoint opertors? The simplest exmple is L d/dx cting on C [, b]. We gin compute Lf, g df b dx gx dg fxdx f, Lg. dx It follows tht L L. Opertors which re their negtive djoints re clled skew symmetric. 2.6 Eigenvlue problems for differentil opertors Anlogous to the mtrix cse, we cn consider n eigenvlue problem for liner opertor L, which sks to find nontrivil functionnumber pirs vx, λ which solve Lvx λvx. 12 It should be noted tht sometimes the eigenvlue problem is written insted like Lvx+λvx which reverses the sign of λ, but the theory ll goes through just the sme. The functions vx which stisfy this re clled eigenfunctions nd ech corresponds to n eigenvlue λ. As with eigenvectors, we cn rescle eigenfunctions: if vx, λ is n eigenvectorvlue pir, so is cvx, λ for ny sclr c. Wht cn we expect to hppen in these problems? Since we re working in infinite dimensions, there re often n infinite number of distinct eigenvlues nd eigenfunctions. The set of eigenvlues is lso clled the discrete spectrum there is nother prt clled the essentil or continuous spectrum which we will not get into. Often the eigenfunctions form bsis for the underlying vector spce, nd the set is is clled complete. This property is extremely vluble in PDEs, since it llows for solutions to be expressed s liner combintions of eigenfunctions. The most fmous exmple of this is the Fourier series, which we discuss lter. Selfdjoint opertors hve some properties equivlent to selfdjoint mtrices. In prticulr, their eigenvlues re rel, nd their eigenfunctions re orthogonl. Orthogonlity is tken with respect to the sme inner product which gives selfdjointness. 2.7 SturmLiouville eigenvlue problems SturmLiouville opertors 9 on C [, b] hve some other nice properties side from those of ny selfdjoint opertor. For the eigenvlue problem the following properties hold: d dx px dv dx + qxvx + λvx, 1. The rel eigenvlues cn be ordered λ 1 < λ 2 < λ 3... so tht there is smllest but not lrgest eigenvlue. 2. The eigenfunctions v n x corresponding to ech eigenvlue λ n form complete set, i.e. for ny f C [, b], we cn write f s infinite liner combintion f c n v n x. 13 n1 6
7 The infinite series in 13 should cuse some concern. It should be noted tht for functions, there re different wys to define convergence. One wy is just to fix ech point x nd consider the limit of function vlues, which is clled pointwise convergence. We ctully hve something stronger here: the rte t which the sequences t ll x converge is the sme. This is clled uniform convergence. Here is the simplest exmple of SturmLiouville eigenvlue problem. Consider the opertor L d 2 /dx 2 on the vector spce C [, π]. The eigenvlue problem reds d 2 v + λv, v, vπ. 14 dx2 This is just second order differentil eqution, nd writing down the generl solution is esy. Recll tht we guess solutions of the form vx exprx. Provided λ >, we get r ±i λ, which mens tht the generl solution hs the form vx A cos λx + B sin λx. Not ll of these solutions re vlid; we require tht v vπ. Therefore A cos+b sin, so tht A. The other boundry condition implies B sinπ λ which is only true if π λ is multiple of π. Therefore λ n n 2, n 1, 2, 3,.... Corresponding to ech of these eigenvlues is the eigenfunction v n x sinnx, n 1, 2, 3,.... Recll tht we don t cre bout the prefctor B since eigenfunctions cn lwys be rescled. Properly speking, we lso need to consider the cses λ, λ <. For the first cse, vx Ax + B, nd it s obvious one needs A B to stisfy the boundry conditions. If λ <, vx A exp λ x + B exp λ x. The boundry conditions imply A + B nd A exp λ π + B exp λ π, which written in mtrix form is 1 1 exp λ π exp λ π A B Such system only hs nonzero solutions if the mtrix is singulr, which it is not since its determinnt is exp λ π exp λ π. Be creful: there re SturmLiouville eigenvlue problems which do hve nonpositive eigenvlues. On the other hnd, there is no need to worry bout complex eigenvlues becuse the liner opertor is selfdjoint. 2.8 Fourier series There is big pyoff from fct tht the eigenfunctions of the previous exmple re complete: we cn write ny function in C [, π] s liner combintion of the eigenfunctions of d2 /dx 2. In other words we cn write ny smooth function with zero boundry vlues s fx. B n sinnx. 15 This is the fmous Fourier sine series  just one of severl types of Fourier series. n1 7
8 Series type Spce of functions Orthogonl expnsion for fx Coefficients Fourier Sine Cosine Complex fx : [ L, L] R f L fl f L f L fx : [, L] R f fl fx : [, L] R f f L fx : [ L, L] C f L fl f L f L A 2 + n1 A n cos nπx + n1 B n sin nπx n1 B n sin nπx L L L A 2 + n1 A n cos nπx L A n 1 L B n 1 L B n 2 L A n 2 L L L L L n c n exp inπx L c n 1 L 2L L Tble 1: Vrious Fourier series nπx fx cos L dx nπx fx sin L dx L nπx fx sin L dx L nπx fx cos L dx inπx fx exp L dx The question tht rises is, how do we ctully compute the coefficients B n? We hve lredy nswered this question in more generl setting. Becuse the eigenfunctions sinnx re orthogonl, we cn use 7 nd 8. For the present cse, this is equivlent to tking n inner product of 15 with ech eigenfunction sinnx. This gives the equivlent of 7, nmely B n fx, sinnx sinnx, sinnx π fx sinnxdx π sin2 nxdx. It should be emphsized tht Fourier coefficient formuls like this one don t need to be memorized. They rise quite simply from the bsic ide of finding coefficients of liner combintion of orthogonl vectors. There re other types of Fourier series involving cosines or complex exponentils. These functions re ll eigenfunctions of the second derivtive opertor L d 2 /dx 2, but with different vector spces of functions on which it cts remember the technicl point erlier: properties of differentil opertors crucilly depend on the vector spce in question. Tble 2.8 summrizes the stndrd types of Fourier series. The coefficient formuls ll derive from tking inner products with ech eigenfunction note tht for the complex series, the inner product is u, v L L uvdx. We shll see lter tht other eigenvlue problems give rise to new orthogonl sets of functions: the Bessel functions, nd the sphericl hrmonics. 2.9 Integrl opertors Another importnt type of liner opertor involves integrls, such s the HilbertSchmidt clss of integrl opertors Lux kx, yuxdx. 16 Notice tht the output of this opertion is function of y, not x. Here is the domin of u nd kx, y is clled the kernel, which is function from to R. Linerity rises simply becuse of linerity of the integrl: Lc 1 u 1 + c 2 u 2 c 1 kx, yu 1 xdx + c 2 kx, yu 2 xdx c 1 Lu 1 + c 2 Lu 2. 8
9 We will not encounter mny equtions which hve integrl opertors, but some of our solutions will involve integrl opertors. This is becuse integrl opertors re often inverses of differentil opertors. This mens tht the inhomogeneous eqution Lu g hs solution u L 1 g where L 1 is n integrl opertor. Integrl opertors hve djoints. For exmple, using the L 2 inner product, the opertor given in 16 stisfies Lu, v vy ux where L is nother integrl opertor L vx kx, yux dxdy kx, yvy dydx u, L v ky, xvxdx. This is not the sme s L; the inputs to the kernel ky, x hve been reversed. 9
MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationSturmLiouville operators have form (given p(x) > 0, q(x)) + q(x), (notation means Lf = (pf ) + qf ) dx
SturmLiouville operators SturmLiouville operators have form (given p(x) > 0, q(x)) L = d dx ( p(x) d ) + q(x), (notation means Lf = (pf ) + qf ) dx SturmLiouville operators SturmLiouville operators
More informationSTURMLIOUVILLE BOUNDARY VALUE PROBLEMS
STURMLIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationSturmLiouville Eigenvalue problem: Let p(x) > 0, q(x) 0, r(x) 0 in I = (a, b). Here we assume b > a. Let X C 2 1
Ch.4. INTEGRAL EQUATIONS AND GREEN S FUNCTIONS Ronld B Guenther nd John W Lee, Prtil Differentil Equtions of Mthemticl Physics nd Integrl Equtions. Hildebrnd, Methods of Applied Mthemtics, second edition
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationMath Theory of Partial Differential Equations Lecture 29: SturmLiouville eigenvalue problems (continued).
Mth 412501 Theory of Prtil Differentil Equtions Lecture 29: SturmLiouville eigenvlue problems (continued). Regulr SturmLiouville eigenvlue problem: d ( p dφ ) + qφ + λσφ = 0 ( < x < b), dx dx β 1 φ()
More informationVariational Techniques for SturmLiouville Eigenvalue Problems
Vritionl Techniques for SturmLiouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More information1 E3102: a study guide and review, Version 1.0
1 E3102: study guide nd review, Version 1.0 Here is list of subjects tht I think we ve covered in clss (your milege my vry). If you understnd nd cn do the bsic problems in this guide you should be in very
More informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
More informationc n φ n (x), 0 < x < L, (1) n=1
SECTION : Fourier Series. MATH4. In section 4, we will study method clled Seprtion of Vribles for finding exct solutions to certin clss of prtil differentil equtions (PDEs. To do this, it will be necessry
More informationChapter 28. Fourier Series An Eigenvalue Problem.
Chpter 28 Fourier Series Every time I close my eyes The noise inside me mplifies I cn t escpe I relive every moment of the dy Every misstep I hve mde Finds wy it cn invde My every thought And this is why
More information1 1D heat and wave equations on a finite interval
1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
More information21.6 Green Functions for First Order Equations
21.6 Green Functions for First Order Equtions Consider the first order inhomogeneous eqution subject to homogeneous initil condition, B[y] y() = 0. The Green function G( ξ) is defined s the solution to
More information1 2D Second Order Equations: Separation of Variables
Chpter 12 PDEs in Rectngles 1 2D Second Order Equtions: Seprtion of Vribles 1. A second order liner prtil differentil eqution in two vribles x nd y is A 2 u x + B 2 u 2 x y + C 2 u y + D u 2 x + E u +
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous relvlued function on I), nd let L 1 (I) denote the completion
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More information2 Sturm Liouville Theory
2 Sturm Liouville Theory So fr, we ve exmined the Fourier decomposition of functions defined on some intervl (often scled to be from π to π). We viewed this expnsion s n infinite dimensionl nlogue of expnding
More informationSturmLiouville Theory
LECTURE 1 SturmLiouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More information3 Mathematics of the Poisson Equation
3 Mthemtics of the Poisson Eqution 3. Green functions nd the Poisson eqution () The Dirichlet Green function stisfies the Poisson eqution with deltfunction chrge 2 G D (r, r o ) = δ 3 (r r o ) (3.) nd
More information4 SturmLiouville Boundary Value Problems
4 SturmLiouville Boundry Vlue Problems We hve seen tht trigonometric functions nd specil functions re the solutions of differentil equtions. These solutions give orthogonl sets of functions which cn be
More information1 E3102: A study guide and review, Version 1.2
1 E3102: A study guide nd review, Version 1.2 Here is list of subjects tht I think we ve covered in clss (your milege my vry). If you understnd nd cn do the bsic problems in this guide you should be in
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for OneDimensionl Eqution The reen s function provides complete solution to boundry
More informationDEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.
398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More informationMapping the delta function and other Radon measures
Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationApplied Partial Differential Equations with Fourier Series and Boundary Value Problems 5th Edition Richard Haberman
Applied Prtil Differentil Equtions with Fourier Series nd Boundry Vlue Problems 5th Edition Richrd Hbermn Person Eduction Limited Edinburgh Gte Hrlow Essex CM20 2JE Englnd nd Associted Compnies throughout
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a nonconstant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More information(9) P (x)u + Q(x)u + R(x)u =0
STURMLIOUVILLE THEORY 7 2. Second order liner ordinry differentil equtions 2.1. Recll some sic results. A second order liner ordinry differentil eqution (ODE) hs the form (9) P (x)u + Q(x)u + R(x)u =0
More informationReview SOLUTIONS: Exam 2
Review SOUTIONS: Exm. True or Flse? (And give short nswer ( If f(x is piecewise smooth on [, ], we cn find series representtion using either sine or cosine series. SOUTION: TRUE. If we use sine series,
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationMATRICES AND VECTORS SPACE
MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN SPACE AND SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationGreen function and Eigenfunctions
Green function nd Eigenfunctions Let L e regulr SturmLiouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationLecture 2e Orthogonal Complement (pages )
Lecture 2e Orthogonl Complement (pges ) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More information221B Lecture Notes WKB Method
Clssicl Limit B Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using
More informationMath 61CM  Solutions to homework 9
Mth 61CM  Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationLECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for
ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.
More informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationSTURMLIOUVILLE THEORY, VARIATIONAL APPROACH
STURMLIOUVILLE THEORY, VARIATIONAL APPROACH XIAOBIAO LIN. Qudrtic functionl nd the EulerJcobi Eqution The purpose of this note is to study the SturmLiouville problem. We use the vritionl problem s
More information1 Line Integrals in Plane.
MA213 thye Brief Notes on hpter 16. 1 Line Integrls in Plne. 1.1 Introduction. 1.1.1 urves. A piece of smooth curve is ssumed to be given by vector vlued position function P (t) (or r(t) ) s the prmeter
More information1. GaussJacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. GussJcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L13. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationSeparation of Variables in Linear PDE
Seprtion of Vribles in Liner PDE Now we pply the theory of Hilbert spces to liner differentil equtions with prtil derivtives (PDE). We strt with prticulr exmple, the onedimensionl (1D) wve eqution 2 u
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationChapter 3 Polynomials
Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationContinuous Quantum Systems
Chpter 8 Continuous Quntum Systems 8.1 The wvefunction So fr, we hve been tlking bout finite dimensionl Hilbert spces: if our system hs k qubits, then our Hilbert spce hs n dimensions, nd is equivlent
More informationNotes on the Eigenfunction Method for solving differential equations
Notes on the Eigenfunction Metho for solving ifferentil equtions Reminer: WereconsieringtheinfiniteimensionlHilbertspceL 2 ([, b] of ll squreintegrble functions over the intervl [, b] (ie, b f(x 2
More informationREPRESENTATION THEORY OF PSL 2 (q)
REPRESENTATION THEORY OF PSL (q) YAQIAO LI Following re notes from book [1]. The im is to show the qusirndomness of PSL (q), i.e., the group hs no low dimensionl representtion. 1. Representtion Theory
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationSTUDY GUIDE FOR BASIC EXAM
STUDY GUIDE FOR BASIC EXAM BRYON ARAGAM This is prtil list of theorems tht frequently show up on the bsic exm. In mny cses, you my be sked to directly prove one of these theorems or these vrints. There
More informationSTURMLIOUVILLE PROBLEMS
STURMLIOUVILLE PROBLEMS Mrch 8, 24 We hve seen tht in the process of solving certin liner evolution equtions such s the het or wve equtions we re led in very nturl wy to n eigenvlue problem for second
More informationThe Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5
The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixedpoint itertion to converge when solving the eqution
More information