PH 610: Mathematical Methods for Scientists

Fall 2022 (D. Belitz)

Chapter 1: Algebraic Structures

$1: Sets, and mappings
1.1 Sets
1.2 Mappings
1.3 Ordered sets
1.4 The natural numbers, and the principle of mathematical induction


$2: Groups
2.1 The definition of a group
2.2 Rules of operation
2.3 Permutations
2.4 Subgroups
2.5 Isomorphisms and authomorphisms


$3: Fields
3.1 Bilinear mappings
3.2 Fields
3.3 The field of complex numbers


$4: Vector spaces and tensor spaces
4.1 Vector spaces
4.2 Basis sets
4.3 Tensor spaces
4.4 Dual spaces
4.5 Metric spaces
4.6 Banach spaces
4.7 Hilbert spaces
4.8 Generalized metrics, Minkowski space

$5: Tensor fields
5.1 Tensor fields
5.2 Gradient, curl, divergence
5.3 Tensor products and tensor traces
5.4 Minkowski tensors

Chapter 2: Topics in Analysis

$1: Reminder: Real analysis
1.1 Differentiation and integration
1.2 Paths, and line integrals
1.3 Surfaces, and surface integrals

$2: Complex-valued functions of one complex variable
2.1 Complex functions
2.2 Analyticity

$3: Integration in the complex plane
3.1 Path integrals
3.2 Laurent series
3.3 The residue theorem
3.4 Simple applications of the residue theorem
3.5 Another application of complex analysis: The Airy function Ai(x)


$4: Fourier transforms, and generalized functions
4.1 The Fourier transform in classical analysis
4.2 Inverse Fourier transforms
4.3 Test functions
4.4 Generalized functions
4.5 The delta function