Syllabus
Vectors and tensors.
Affine space, vectors and linear forms, tensors, coordinate transformations, diagrammatic notation, scalar product and metric.
Curvilinear coordinates and vector analysis.
Tensor fields, gradient and nabla-operator, curvilinear coordinates, triads. Integrating vectors and tensors.
Introduction to distributions.
Basic definitions and properties, δ-distribution, derivatives of non-smooth functions, regularization of 1/x. Fourier transformation of distribution, examples. Distribution on manifolds, characteristic function, surface and linear δ-distributions and their derivatives. Aplications: point, linear and surface sources, dipoles, boundary conditions for electrostatic a magnetostatic, electric field near conductors.
Green functions
Green functions in one variable. Green function for Laplace operator, Laplace equation on a domain with a boundary, heat equation, perturbative solution of Schrödinger equation with potential.
Classical field theory
Lagrange and Hamilton formalism for fields, scalar and electromagnetic field, gauge symmetry.
Supplements for classical electrodynamics
Multipole expansion in terms of tensors. Description of continuum is spacetime, stress-energy tensor, electric current density, conservation laws.
From sum over trajectories to solution of differential equations.
Feynman's formulation of quantum mechanics: quantum histories, quantum indistinguishability, amplitude rules, measurement model. Path integral, amplitude of free particle evolution, perturbative solution of Schrödinger equation.
Annotation
Vector and tensor calculus, curvilinear coordinates. Introduction to distributions, Fourier transformation, Green functions.
Introduction to classical field theory. Multipole expansion in a tensor form.
Feynman formulation of quantum mechanics.