These topics will be covered here, and the reading materials can be accessed by clicking on the hyperlinks.

1. Vector Algebra & Calculus
   • Vector algebra: addition, scalar and vector products, triple products
   • Gradient, divergence, and curl — physical interpretation
   • Line, surface, and volume integrals
   • Integral theorems: Gauss, Green, and Stokes
   • Scalar and vector potentials; Laplacian operator
   • Curvilinear coordinates (cylindrical and spherical) and scale factors
2. Linear Algebra
   • Vector spaces, linear dependence and independence
   • Basis and dimension, subspaces
   • Matrices and determinants, adjoint and inverse
   • Rank of a matrix and linear transformations
   • Eigenvalues and eigenvectors; characteristic equation
   • Hermitian, skew-Hermitian, unitary, and orthogonal matrices
   • Diagonalization and spectral theorem
3. Ordinary Differential Equations (ODE) & Special Functions
   • First and second order differential equations
   • Linear differential equations with constant coefficients
   • Series solutions — Frobenius method
   • Legendre, Bessel, Hermite, and Laguerre equations and their properties
   • Orthogonality of functions
   • Sturm–Liouville problems
4. Computational Techniques
   • Numerical methods: root finding (bisection, Newton–Raphson)
   • Numerical differentiation and integration (Trapezoidal, Simpson)
   • Runge–Kutta methods for ODEs
   • Error analysis and propagation
   • Finite difference methods
   • Basics of programming for physics simulations
5. Thermodynamics
   • Laws of thermodynamics and their applications
   • Thermodynamic systems, variables, and equations of state
   • Maxwell’s relations and thermodynamic potentials
   • Clausius–Clapeyron equation
   • Joule–Thomson effect, adiabatic and isothermal processes
   • Thermodynamic equilibrium and stability
6. Statistical Physics
   • Microstates and macrostates, ensembles (microcanonical, canonical, grand canonical)
   • Boltzmann distribution and partition function
   • Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics
   • Classical and quantum limits of ideal gases
   • Blackbody radiation and Planck’s distribution
   • Fluctuations, equipartition theorem, and entropy

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📚 Reference Books

• Mathematical Methods for Physicists — Arfken & Weber
• Linear Algebra and Its Applications — Gilbert Strang
• Thermal Physics — Zemansky & Dittman
• Statistical Mechanics — Pathria & Beale
• Mathematical Physics — B. S. Rajput / H. K. Dass