PH: 301 Methods of Mathematical Physics 

  1. Review of vector analysis: definitions, rotation of coordinate axes, scalar product, cross product, addition of vectors.
  2. Differential operators, gradient, divergence, curl, integration of vector fields, Gauss' theorem, Stokes' theorem, Gauss' law, Poisson's equation.
  3. Definition of delta function, representations including plane wave expansion, generalization to 3 dimensions.
  4. Vector analysis in curvilinear coordinates, orthogonal coordinates in R3, circular and spherical coordinates, definition of tensors, contraction, direct product, quotient rule, pseudo tensors, dual tensors, tensor derivative operators.
  5. Determinants, matrices, orthogonal and unitary matrices, matrix diagonalization, trace theorem, relation between determinants and traces.
  6. Finite and infinite sequences, limit of a sequence.
  7. Finite and infinite series, tests of convergence, alternating series, algebra of series, series of functions, Taylor's expansion and power series, Bernoulli numbers, Euler-Maclaurin formula, asymptotic series, infinite products
  8. Fourier series and analysis, use and application to physical systems. orthogonality and orthonormality, complete sets of functions, Gibbs phenomenon, discrete and continuous Fourier transform.
  9. Complex algebra, functions of a complex variable, Cauchy-Riemann conditions, integration of complex functions, calculus of residues, Cauchy's theorem, Laurent expansion, dispersion relations.


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