Mathematical Methods MT 2005: F.H.L. Essler

The course comprises of 20 lectures: three per week in weeks 1-4 and two per week in weeks 5-8.
The course is divided into five parts, each of which is supported by a problem set. A collection paper for
week 0 of HT 2006 will be available to tutors.

Part I: Linear Algebra I [Recommended Reading] [Notes][Problem Set I]
Euclidean Linear Vector Spaces; Real vs Complex Vector Spaces; Dual Vectors and Scalar Product;
Linear Independence; Dimension; Bases;

Part II: Linear Algebra II [Recommended Reading] [Notes] [Problem Set II]
Linear Operators; Matrices; Matrix Algebra; Rotations; Lorentz Transformations; Special Operators;
Eigenvalues; Eigenvectors; Orthogonalization; Change of Basis; Orthonormal Bases; Invariant Subspaces;
Hermitian Matrices; Diagonalization of Hermitian Matrices;

Part III: Fourier Methods [Recommended Reading] [Notes] [Problem Set III]
Fourier Series; Fourier Transforms as Limit of Fourier Series; Inverse Transform; Dirac Delta Function;
Distributions; Parseval's Theorem; Convolution; Applications;

Part IV: Partial Differential Equations [Recommended Reading] [Notes] [Problem Set IV]
Examples; Initial Conditions; Boundary Conditions; Separation of Variables; Use of Cartesian,
Spherical Polar and Cylinder Coordinates;

Part V: Ordinary Differential Equations [Recommended Reading] [Notes] [Problem Set V]
Difference Equations; Differential Equations as limits of Matrix Equations; Boundary Conditions and
Eigenvalues; Green's Functions; Second order ODEs of Sturm-Liouville Type; Orthogonality of
Eigenfunctions; Legendre's Equation; Hermite's Equation; Eigenfunction Expansions;