Differential Geometry (and Relativity)

Lecture Notes on Mathematical Statistical Physics

This note explains the following topics: classical statistical mechanics, Review of classical mechanics, Review of probability and measure, The Maxwellian distribution Probability spaces in classical mechanics, Review of thermodynamics Macro states, Macro variables, Thermal equilibrium and entropy, The Boltzmann equation, The thermodynamic arrow of time, Quantum statistical mechanics and thermodynamic ensembles.

Author(s): Roderich Tumulka

173 Pages

Mathematical Physics by Indu Satija

This note covers Laws of nature and mathematical beauty, Gaussian Integrals and related functions, Basic gaussian integrals, Stirling formula error functions, Real numbers, Complex numbers, Scalars, Vectors, Tensors and spinor, Fourier transformation, Curvilinear coordinates, Partial differential equations, Solving partial differential equation by separation of variables, Solving laplace equation in spherical polar coordinates, Spherical harmonics and legendre functions, Bessel function, Spherical bessel function and matrices.

Author(s): Indu Satija

120 Pages

Mathematical Physics by Michael Aizenman

The PDF covers the following topics related to Mathematical Physics : Introduction to statistical mechanics, Canonical Ensembles for the Lattice Gas, Configurations and ensembles, The equivalence principle, Generalizing Ensemble Analysis to Harder Cases, Concavity and the Legendre transform, Basic concavity results, Concave properties of the Legendre transform, Basic setup for statistical mechanics, Gibbs equilibrium measure, Introduction to the Ising model, Entropy, energy, and free energy, Large deviation theory, Free energy, Basic Properties, Convexity of the pressure and its implications, Large deviation principle for van Hove sequences, 1-D Ising model, Transfer matrix method, Markov chains, 7 2-D Ising model, Ihara graph zeta function, Gibbs states in the infinite volume limit, Conditional expectation, Symmetry and symmetry breaking, Phase transitions, Random field models, Proof of symmetry-breaking of continuous symmetries, The spin-wave perspective, Infrared bound, Reflection positivity.

Author(s): Professor Michael Aizenman

76 Pages

Lecture Notes for Mathematical Methods of Physics

This note covers the following topics: Series of Functions, Binomial Theorem, Series Expansion of Functions, Vectors, Complex Functions, Derivatives, Intergrals, and the Delta Function, Determinants, Matrices, Vector Analysis, Vector Differentiation and Integration, Integral Theorems and Potential Theory, Curvilinear Coordinates, Tensor Analysis, Jacobians and Differential Forms, Vectors in Function Spaces, Gram-Schmidt Orthogonalization and Operators, Transformations, Invariants, and Matrix Eignevalue Problems, Hermitian and Normal Matrix Eigenvalue Paroblems, Ordinary Differential Equations, Second-Order Linear ODEs, Green's Functions.

Author(s): Gregory G. Howes

NA Pages

Maths for Physics

Mathematics is an integral component of all of the scientific disciplines, but for physics, it is a vital and essential skill that anyone who chooses to study this subject must master. Topics covered includes: Functions and Geometry, Complex Numbers, Matrices, Vectors, Limits, Differentiation, Partial Differentiation and Multivariable Differential Calculus, Integration, Multiple Integration, Differential Equations, Series and Expansions, Operators, Mechanics.

Author(s): Daniel Brett and Joseph Vovrosh

263 Pages

Mathematical Physics by Bergfinnur Durhuus and Jan Philip Solovej

The main focus of this note is on theoretical developments rather than elaborating on concrete physical systems, which the students are supposed to encounter in regular physics courses. Topics covered includes: Newtonian Mechanics, Lagrangian Mechanics, Hamiltonian Mechanics, Hilbert Spaces, Operators on Hilbert spaces and Quantum mechanics.

Author(s): Bergfinnur Durhuus and Jan Philip Solovej

177 Pages