Mathematical Methods - DAMTP

1 Mathematical MethodsUniversity of Cambridge Part IB Mathematical TriposDavid SkinnerDepartment of Applied Mathematics and Theoretical physics ,Centre for Mathematical Sciences,Wilberforce Road,Cambridge CB3 0 WAUnited are the lecture notes for the Mathematical Methods course given tostudents taking Part IB Maths in Cambridge during Michaelmas Term of 2015. The courseaims to provide an introduction to Fourier Series and Fourier Transforms, Sturm LiouvilleTheory, and PDEs such as the Laplace, heat and wave Fourier Spaces of functions as infinite dimensional vector Fourier A first look at with rate of Fej er s Parseval s identity172 Sturm Liouville Self-adjoint Differential Self-adjoint differential Eigenfunctions and weight Some Inhomogeneous equations and Green s Parseval s identity Least squares approximation293 Laplace s Laplace s equation on a Separation of The Laplacian in spherical polar Cosmic Microwave s equation on the Laplace s equation in cylindrical polar value problems in cylindrical coordinates474 The Heat The fundamental Heat flow as a smoothing transatlantic Atiyah Singer index theorem55 i Brownian motion and the existence of Boundary conditions

2 And Heat conduction in a plane Steady heat conduction in a finite Cooling of a uniform sphere615 The Wave Vibrations of a uniform Energetics and Vibrations of a circular Can one hear the shape of a drum? Wave reflection and transmission726 Generalized Test functions and of Properties of the Dirac expansion of the Schwartz functions and tempered distributions837 Green s Functions for Ordinary Differential Construction of the Green s Physical interpretation of the Green s Green s functions for inhomogeneous boundary Equivalence of eigenfunction expansion ofG(x; ) Application of Green s functions to initial value Higher order differential operators958 Fourier Simple properties of Fourier The Fourier inversion s theorem for Fourier The Fourier transform of Schwartz functions and tempered transform of the Dirac Linear systems and transfer form of transfer functions for ode The discrete Fourier Fourier transforms1119 The Method of fields and integral for first order Characteristics for second order ii of Alembert s general solution of the wave holes12510 Green s functions for Fourier transforms for the heat The forced heat Duhamel s Fourier transforms for the wave Poisson s The fundamental Green s Dirichlet Green s functions for the Laplace & Poisson The method of Green s function for

3 The Laplace s equation on the half Method of images for the heat Method of images for inhomogeneous problems141 iii PreliminariesRecommended BooksWhile these notes should contain the material we cover in the lectures, they re very farfrom a comprehensive treatment and undoubtedly reflect my idiosyncracies in choice ofmaterial. So it s a good idea to balance them with a proper textbook. Here are some ofthe ones I ve found useful in preparing the course. Arfken, G. and Weber, H., Mathematical Methods for Physicists, Academic (2005).The single most suitable book for this course. Covers all the core material. Boas, M., Mathematical Methods in the Physical Sciences, Wiley (2005).Also appropriate for this course. Mathews, J. and Walker, Methods of physics , Benjamin Cummins(1970). Jeffreys, H. and Jeffreys of Mathematical physics , CUP 3rdedition (1999).

4 A classic. To be found on the shelves of many generations of Mathematical physicists. K orner, Analysis, Cambridge (1989).More advanced, but wonderful. Very engagingly written with a unique blend of math-ematical rigour and historical anecdote. I enjoyed reading this a lot when preparingthese notes. Renardy, M. and Rogers, Introduction to Partial Differential Equations, Springer(2004).Again more advanced; contains lots of extra material going into further depth in thelater parts of this in these lecture notes is original to me. In particular, the notes are based on lecturenotes I inherited from Prof R. Jozsa, which were in turn inherited from Prof. C. can still find Prof. Jozsa s notes here. I ve also borrowed heavily from sections of thebooks listed above. Any errors, major or minor, are of course mine. If you spot one pleaseemail me and point it am supported in part by the European Research Council under an FP7 Marie CurieCareer Integration Grant.

5 Iv 1 Fourier SeriesMany of the most important equations of Mathematical physics arelinear, includingLaplace s equation( 2 x2+ 2 y2) (x,y) = 0 The heat (or diffusion) equation( t K 2 x2) (t,x) = 0 The wave equation(1c2 2 t2 2 x2) (t,x) = 0 Schr odinger s equation(i~ t+~22m 2 x2 V(x)) (t,x) = 0 Maxwell s vacuum equations E= 0 B= 0 E= 1c B t B=1c E tLinearity means that if we are given two solutions 1and 2of one of these equations say the wave equation then 1 1+ 2 2is also a solution for arbitrary constants 1, one possible exception, the real reason all these equations are linear is the same:they re approximations. The most common way for linear equations to arise is by slightlyperturbing a general system. Whatever the complicated equations governing the dynamicsof the underlying theory, if we just look to first order in the small perturbations then we llfind a linear equation essentially by definition1.

6 For example, the wave equation will givea good description of ripples on the surface of a still pond, or light travelling through apane of glass, but don t expect to use it to find out how big a splash you ll make when youbomb into the swimming pool, or if we shine a strong laser at the glass. Similarly, we lllearn how to use the heat equation to tell us about the average jiggling of the atoms in ametal bar when it s being gently warmed somewhere, but if we jiggle them too much thenthe metal bar will possible exception is Schr odinger s equation in Quantum Mechanics. We knowof many ways to generalize this equation, such as making it relativistic or passing toQuantum Field Theory, but in each case the analogue of Schr odinger s equation alwaysremains exactly linear. No one knows if there is a fundamental reason for this (though it scertainly built into the principles of Quantum Mechanics at a deep level), or whether ourexperiments just haven t probed far any case, learning to solve linear differential equations such as the above, andtheir generalizations to higher dimensions, is an important first step in understanding thedynamics of a very wide class of physical (and even biological) systems.

7 Fourier s insightwas to take linearity as the key: if we can find a class of simple solutions then we may beable to construct a more general one by taking linear combinations of with a source term. 1 VectorsLet s begin by recalling a few facts about vectors that you met last year. Avector spaceover a fieldF(in this course we ll always takeF=RorF=C) is defined to be a setVtogether with the operation + of addition, obeyingcommutativityu+v=v+uassociativity u+ (v+w) = (u+v) +widentity !0 V +u=ufor allu,v,w V, and the operation of multiplication by a scalar Fthat isdistributive inV (u+v) = u+ vdistributive inF( + )u= u+ s often useful to give our vector space aninner product. This is a choice of map(,) :V V Fthat obeys2conjugate symmetry(u,v) = (v,u) linearity(u, v) = (u,v)additivity(u,v+w) = (u,v) + (u,w)positive-definiteness(u,u) 0 for allu V, with equality iffu= thatifour vectors are real, then the property (u,v) = (v,u) implies that (,) issymmetric in its arguments.

8 In this case, the map (,) :V V Ris bilinear. IfF=Cthe map is sometimes inner product onV Vautomatically provides our vector space with anorm. Wecan define the length of a vectoruto be the norm (u,u). In the real case, the anglebetween two vectors is given by = arccos((u,v) (u,u) (v,v))( )where the Cauchy-Schwartz inequality (u,v)2 (u,u) (v,v) ensuring that this set of vectors{v1,v2,..,vn}form abasisofVif any elementu Vcan be uniquelywritten asu= ni=1 ivifor some scalars i. Thedimensionof the vector space is thenumber of elements of any basis. A basis{v1,v2,..,vn}isorthogonal wrt the innerproductif (vi,vj) vanishes wheneveri6=j, the name coming from ( ). The basis isorthonormalif also the length of eachviis 1. If we re given an orthonormal basis, we canuse the inner product to explicitly decompose a general into this basis.

9 For example, ifu=n i=1 ivi,( )2 Beware! It s very common for some authors to define the inner product to be linear in thefirstentry,rather than the second as I have done here. I ve chosen this way for maximal agreement with your QuantumMechanics lectures. And because I m a physicist. 2 then we have(vj,u) =(vj,n i=1 ivi)=n i=1(vj, ivi) =n i=1 i(vj,vi) = j,( )where we used the additivity and linearity properties of (,), as well as orthonormality ofthe basis. Thus we find j= (vj,u). For real vectors, jis just the projection Spaces of functions as infinite dimensional vector spacesConsider the set of complex valued functions on some domain . Such a functionfcanbe viewed as a mapf: C. The set of all such functions is naturally thought of as avector space, where vector addition + is just pointwise addition of the functions; that is,forx we have(f+g)(x) =f(x) +g(x)( )where the addition on the rhs is just addition inC.

10 Likewise, we can multiply functionsby scalars as( f)(x) = f(x),( )where again the multiplication is just the usual multiplication about the inner product? One possible choice is to take(f,g) f(x) g(x) d ( )where d is some choice of integration measure, and where the functionsf(x) andg(x)are sufficiently well-behaved that the integral exists. The idea is that this is a simplegeneralization of the inner product between two finite dimensional vectors: if we think ofthe different pointx as labelling the different components of our functions, then wemultiply the component f(x) offand the component g(x) ofgtogether (after taking anappropriate complex conjugate) and then add them up, ,integrate over . The measured tells us how much weight to assign to each point of the a simple example, if is the interval [a,b], then we may take the measure to bejust dxso that(f,g) = baf(x) g(x) dx.