Lecture Notes of PHY 503, Mathematical Methods

Below you find the lecture notes of last year. This year will be mostly the same at least for the beginning of the class.

Lecture 1, August 22
Functionals (1.2), Functional Derivatives (1.2.1), Euler-Lagrange equations (1.2.2), First Integral (1.2.4)

Lecture 3, August 26
Constraints (1.5), Catenary (1.5)

Lecture 4, August 29
>Action principle (1.3), Noether's theorem (1.3.2)

Lecture 5, August 31
Linear Vector spaces (A.1.1), Bases (A.1.2)

Lecture 6, September 2
Linear maps (A.2.1) Range Null-Space Theorem (A.2.2), Dual space (A.2.3),

Lecture 7, September 7
Scalar Product (A.3.1) Orthonormal basis (A.3.1) , Euclidean Vectors (A.3.2), Bra-Ket Notation (A.3.3)

Lecture 8, September 9
Conjugate Map (A.3.3), Adjoint Operator (A.3.4), Direct Sum (A.4.1) , Qoutient space (A.4.2)

Lecture 11, September 16
Rank of a Matrix (A.5.1), Fredholm Alternative (A>5.2)

Lecture 12, September 19
Determinants, skew-symmetric forms (A.6)

Lecture 13, September 21`
Properties of determinants (A.6.1), Adjugate Matrix (A.6.2)

Lecture 14, September 23
Answered questions and discussed examples

Lecture 15, September 26 Derivativ of a Determinant (A.6.3) Characteristic Equation (A.6.2) , Cayley's Theorem (A.6.2)

Lecture 16, September 28
Answered questions and discussed examples

Lecture 17, September 30
Diagonalization (A.7), When is a matrix diagonalizible (A.7.1)

Lecture 18, October 3
Jordan Canonical Form (A.7.1)

Lecture 19, October 5
Quadratic Forms (A.7.2)

Lecture 20, October 7
Function Spaces (2.1), Norm (2.2.1), Convergence (2.2.1)

Lecture 21, October 12
Almost All (2.2.2), Cauchy Sequence (2.2.2), Banach Space (2.2.3), Hilbert Space (2.2.3)

Lecture 22, October 14
Orthogoanal Function Sets (2.2.3)

Lecture 23, October 17
Best Approximation (2.2.3), Parcival's Theorem (2.2.3), Orthogonal Polynomials (2.2.3)

Lecture 24, October 19
Three Step Recursion Relation (2.2.3), Weierstrass approximation theorem (2.2.3)

Lecture 25, October 21
, Construction of Orthogonal Polnomials (2.2.4), Legendre Polynomials (2.2.4)

Lecture 26, October 24
Hermite Polynomial (2.2.4) , Tchebychev Polynomials (2.2.4)

Lecture 27, October 26
Distributions (2.3.1), Test Functions (2.3.2)

Lecture 29, October 31
Weak Derivative (2.3.2), Principe Value Integral (2.3.2)

Lecture 30, November 2
Linear Differential Equations (3.1.1)

Lecture 34, November 11
Normal Form (3.2), Inhomogeneous Differential Equations (3.3)

Lecture 35-36, November 14
Variation of Parameters (3.3.2), Green's Functions (5)

Lecture 36-37, November 16
Green's Function of Sturm-Liouville Operator (5.2.1)

Lecture 37-38, November 18
Green's Function of Initial Value Problem (5.2.2), Force Harmonic Oscillator (5.2.2)

Lecture 39, November 28
Poisson Equation, Forced Damped Harmonic Oscillator (5.2.2)

Lecture 40, November 30
Modified Green's Function (5.2.3), Eigenfunction Expansion of Green's Function (5.4)

Lecture 41, December 2
Causality and Analyticity (5.5.1)

Lecture 42, December 5
Plemelj Formulae (5.5.2), Resolvent (5.5.3)