Physics 215

Syllabus

1. Vectors and Vector Calculus
   • scalar, vector and triple products, permutation symbol
   • vector calculus with gradient, divergence and curl
   • orthogonal curvilinear coordinate systems, examples
   • vector integration, line, surface and volume integrals
   • Gauss's theorem, Gauss's law, delta function
   • Stokes theorem
   • potential theory, Helmholtz's theorem
   • vector calculus in the orthogonal curvilinear coordinate systems
2. Rotations, Matrices, Eigenvalue Problem
   • Vectors, matrices, determinants, systems of linear equations.
   • Matrices as representation of linear operator.
   • Changes of base and unitary matrixes.
   • Eigenvalues and eigenvectors of a (Hermitian) matrix; diagonalization of a matrix, problem of eigenvectors.
   • rotations in space, angular momentum, (elements of group theory)
   • (tensor notation )
   • examples from classical mechanics and electrodynamics.
3. Differential Equations
   • first order differential equations, exact equations, inhomogenous first order differential equations
   • second order differential equations, homogenous and inhomogenous second order differential equations
   • Green's function
   • power series, Frobenius method, a second solution
   • Partial differential equations of physics, separation of variables, boundary value problems,separation of variables for the Laplacian
4. Special functions and Generalized Fourier Series
   • Legendre function and spherical harmonics
   • Bessel functions of first type
   • Bessel functions of second type. Neumann functions
   • Hermite functions
   • Problem of Sturm-Liouville, self-adjoint operators, eigenvalues and eigenfunctions
   • eigenfunction expansion and (generalized) Fourier series
   • Dirac notation and projection operators. Orthogonal polynomials. Structure of Quantum Mechanics, examples