Methods of Applied Mathematics Lecture Notes

1 Methods of Applied MathematicsLecture NotesWilliam G. FarisMay 14, 20022 Contents1 Linear Matrices .. algebra .. row echelon form .. Jordan form .. forms .. theorem .. (convolution) matrices .. Vector spaces .. spaces .. transformations .. row echelon form .. form .. and dual spaces .. forms .. relativity .. products and adjoint .. theorem .. Vector fields and differential forms .. systems .. fields .. forms .. of a vector field near a zero .. approximation to a function at a critical , gradient, and divergence .. polar coordinates .. systems .. systems .. Contravariant and covariant.

2 Problems .. 3434 CONTENTS2 Fourier Orthonormal families .. Absolute convergence .. Pointwise convergence .. Problems .. 423 Fourier Introduction .. Absolute convergence .. Fourier transform pairs .. Problems .. Poisson summation formula .. Problems .. PDE Problems .. 514 Complex Complex number quiz .. Complex functions .. and exact forms .. equations .. Cauchy integral theorem .. representation .. cuts .. Complex integration and residue calculus .. Cauchy integral formula .. residue calculus .. residue calculation .. Problems .. More residue calculus.

3 S lemma .. more delicate residue calculation .. formula for derivatives .. of higher order .. residue calculation with a double pole .. The Taylor expansion .. of convergence .. surfaces .. 64 CONTENTS55 Properties of distributions .. Mapping distributions .. Radon measures .. Approximate delta functions .. Problems .. Tempered distributions .. Poisson equation .. Diffusion equation .. Wave equation .. Homogeneous solutions of the wave equation .. Problems .. Answers to first two problems .. 796 Bounded Introduction .. Bounded linear operators .. Compact operators.

4 Hilbert-Schmidt operators .. Problems .. Finite rank operators .. Problems .. 907 Densely Defined Closed Introduction .. Subspaces .. Graphs .. Operators .. The spectrum .. Spectra of inverse operators .. Problems .. Self-adjoint operators .. First order differential operators with a bounded interval: pointspectrum .. Spectral projection and reduced resolvent .. Generating second-order self-adjoint operators .. First order differential operators with a semi-infinite interval:residual spectrum .. First order differential operators with an infinite interval: contin-uous spectrum .. Problems.

5 A pathological example .. 1046 CONTENTS8 Normal Spectrum of a normal operator .. Problems .. Variation of parameters and Green s functions .. Second order differential operators with a bounded interval: pointspectrum .. Second order differential operators with a semibounded interval:continuous spectrum .. Second order differential operators with an infinite interval: con-tinuous spectrum .. The spectral theorem for normal operators .. Examples: compact normal operators .. Examples: translation invariant operators and the Fourier trans-form .. Examples: Schr odinger operators .. Subnormal operators .. Examples: forward translation invariant operators and the Laplacetransform.

6 Quantum mechanics .. Problems .. 1199 Calculus of The Euler-Lagrange equation .. A conservation law .. Second variation .. Interlude: The Legendre transform .. Lagrangian mechanics .. Hamiltonian mechanics .. Kinetic and potential energy .. Problems .. The path integral .. Appendix: Lagrange multipliers .. 13110 Perturbation The implicit function theorem: scalar case .. Problems .. The implicit function theorem: systems .. Nonlinear differential equations .. A singular perturbation example .. Eigenvalues and eigenvectors .. The self-adjoint case .. The anharmonic oscillator .. 142 Chapter 1 Linear Matrix algebraAnmbynmatrixAis an array of complex numbersAijfor 1 i mand1 j vector space operations are the sumA+Band the scalar the same dimensions.

7 The operations are defined by(A+B)ij=Aij+Bij( )and(cA)ij=cAij.( )Thembynzero matrix is defined by0ij= 0.( )A matrix is a linear combination of other matrices if it is obtained from thosematrices by adding scalar multiples of those anmbynmatrix andBbe annbypmatrix. Then the productABis defined by(AB)ik=n j=1 AijBjk.( )Thenbynidentity matrix is defined byIij= ij.( )Here ijis the Kronecker delta that is equal to 1 wheni=jand to 0 wheni6= algebra satisfies the usual properties of addition and many of theusual properties of multiplication. In particular, we have the associative law(AB)C=A(BC)( )78 CHAPTER 1. LINEAR ALGEBRAand the unit lawAI=IA=A.( )Even more important, we have the distributive lawsA(B+C) =AB+AC(B+C)A=BA+CA.( )However multiplication is not commutative; in generalAB6= invertible if there is anothernbynmatrixBwithAB=IandBA=I.

8 In this case we writeB=A 1. (It turns out that ifBA=I, then alsoAB=I, but this is not obvious at first.) The inverse operation hasseveral nice properties, including (A 1) 1=Aand (AB) 1=B 1A notion of division is ambiguous. SupposeBis invertible. Then bothAB 1andB 1 Aexist, but they are usually not annbynsquare matrix. The trace ofAis the sum of the diagonalentries. It is easy to check that tr(AB) = tr(BA) for all such matrices. Althoughmatrices do not commute, their traces Reduced row echelon formAnmcomponent vector is anmby 1 matrix. Theith standard basis vector isthe vector with 1 in theith row and zeros everywhere in reduced row echelon form (rref) if each columnis either the next unit basis vector, or a a linear combination of the previousunit basis vectors.

9 The columns where unit basis vectors occur are called pivotcolumns. The rankrofRis the number of pivot IfAis anmbynmatrix, then there is anmbymmatrixEthatis invertible and such thatEA=R,( )whereRis in reduced row echelon form. The matrixRis uniquely theorem allows us to speak of the pivot columns ofAand the rank ofA. Notice that ifAisnbynand had rankn, thenRis the identity matrix andEis the inverse LetA= 412216100312 2 1 3 0 2 3 .( )Then the rref ofAisR= 1 3 0 2 00 0 1 4 00 0 0 0 1 .( )Corollary. LetAhave reduced row echelon formR. The null space ofAisthe null space ofR. That is, the solutions of the homogeneous equationAx= 0are the same as the solutions of the homogeneous equationRx= MATRICES9 Introduce a definition: A matrix is flipped rref if when flipped left-right(fliplr) and flipped up-down (flipud) it is way reduced row echelon form is usually defined, one works from leftto right and from top to bottom.

10 If you try to define a corresponding conceptwhere you work from right to left and from bottom to top, a perhaps sensiblename for this is flipped reduced row echelon LetAbe anmbynmatrix with rankr. Then there is a uniquenbyn rmatrixNsuch that the transpose ofNis flipped rref and such thatthe transpose has pivot columns that are the non pivot columns ofAand suchthatAN= 0( )The columns ofNare called the rational basis for the null space ofA. It iseasy to findNby solvingRN= 0. The rational null space matrixNhas theproperty that its transpose is in flipped reduced row echelon : In the above example the null space matrix ofAisN= 3 2100 40100 .( )That is, the solutions ofAx=0are the vectors of the formx=Nz. In otherwords, the columns ofNspan the null space can also use the technique to solve inhomogeneous equationsAx= simply applies the theory to the augmented matrix [A b].