The
course
comprises of 20 lectures: four per week in weeks 1-4 and two per week
in
weeks 5-6.The
course
is divided into five parts, each of which is supported by a problem
set.
A collection paper for week
0 of HT 2010 will be available to tutors.
Part
I: Linear
Algebra I[Recommended Reading][Problem Set I]
Euclidean Linear Vector Spaces; Real vs Complex Vector Spaces; Dual
Vectors and Scalar Product; Linear Independence; Dimension; Bases;
Different Bases and Orthogonality;
Part
II: Linear
Algebra II[Recommended Reading][Problem Set II]
Linear
Operators;Matrices; Commutator; Functions of Operators; Matrix
Representations of Linear Operators; Operations on Square Matrices;
Change of Basis; Unitary and OrthogonalTransformations; Eigenvalues and
Eigenvectors;
Hermitian Matrices; Diagonalization of Hermitian Matrices; Jordan Normal Form; Simultaneous
Diagonalization of Hermitian Matrices; Tensor Product of Vector Spaces;
Part III: Fourier Methods and Generalized Functions[Recommended Reading][Problem
Set III] Fourier Series; Fourier
Transforms
as Limit of Fourier Series; Inverse Transform; Dirac Delta Function;
Parseval's Theorem; Convolution;
Part IV: Ordinary
Differential Equations[Recommended Reading]
[Problem
Set IV]
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;
Part V: Partial Differential
Equations[Recommended
Reading][Problem Set V]
Examples; Initial
Conditions; Boundary
Conditions; Separation of
Variables;
Use of Cartesian, Spherical Polar and Cylinder
Coordinates;