(one-semester course)

*Prof. Lipovka A.A.*

**Tensors. (2 weeks)**- Definitions
- Properties.
- Fundamental tensor. Examples for curved spaces.
- Co- and contra-variant tensors. Geometric interpretation.
- Dual tensors, invariant values.
- Geodesic lines.
- Christoffel symbols.
- Parallel shift and covariant derivative.
- Contravariant derivative. Equations for geodesic line.
- Generalization of differential operators to the case of N-space.
- Curvature tensor (the tensor of Riemann – Christoffel).
- Identity of Bianchi.
- Ricci tensor, Ricci scalar, Einstein tensor.
- Problem solving, midterm exam.

**The calculus of variations based on the tensors formalism. (2 weeks)**- Introduction
- The brachistochrone problem.
- Euler’s method.
- Classical mechanics, the laws of conservation. Lagrange function.
- Action for a free particle. Lagrange function.
- Action for a charged particle in the electromagnetic field. Lagrange function.
- Derivative of Lagrange – Euler.
- Field equations. Lagrangian function, the energy – momentum tensor.
- Examples (method applied to scalar, vector, tensor fields).
- Problem solving, midterm exam.

**Typical problems in physics, based on 1-st and 2-nd parts. (1 week)**- Differential operators in curved coordinates. Lamé coefficients as components of the metric tensor, operators Δ and ▼ in curved spaces
- Equation of vibrations for a string, membrane, bar. (deduction)
- Equation of diffusion and heat flow. Border conditions. (deduction)
- Equation of the long line (telegraph equation). (deduction)
- Electrostatic equation. (deduction)
- Ideal liquid dynamics equations. (deduction)
- Equations of the elasticity theory. (deduction)
- Classification of differential equations.
- Dirichlet problem for Poisson equation, conditions for unique solution.
- Neumann’s problem for Poisson’s equation, conditions for a unique solution.
- Problem with type III border conditions.
- Laplace’s external problem.
- Boundary conditions of IV-th type. Typical problems.

**Methods of solution. (2 weeks)**- The general solution structure for the partial differential equation.
- Problem of an infinite string (Method of d’Alembert).
- Separation of variables (Fourier method).
- Infinite plate cooling problem. Evaluation of convergence of the solution.
- Problem of a finite string.
- Dirichlet’s problem for a circle. Solution with the Poisson integral.
- External Dirichlet problem for a circle.
- Frobenius method.
- Example 1 – Legendre equation.
- Example 2 – Bessel equation.

**Sturm-Liouville problem (2 weeks)**- Hilbert space, its properties, operators, set of orthogonal functions.
- Gram-Schmidt orthogonalization process. Example: Legendre polynomials.
- Definition of SLP, operator properties.
- Sturm-Liouville problem regular inside the boundaries.
- Properties of eigenvalues and eigenfunctions.
- Lower boundedness of the spectrum of the Sturm-Liouville problem (SLP).
- Determination of the eigenvalues and eigenfunctions of the SLP.
- SLP. Relationship between the norm of eigenfunctions and the characteristic equation.
- Regular SLP with periodic conditions at the border. The foundations of the appearance of quantization in quantum mechanics.
- Singular Sturm-Liouville problem. Properties of eigenvalues and eigenfunctions.
- General scheme for the Fourier method in the case on N variables.
- Example 1 – Thermal flow.
- Example 2 – problem for spherical shell.
- Problem solving, midterm exam.

**Non-homogeneous problems. Based on parts 3,4,5. (2 weeks)**- Reduction to the homogeneous problem. General scheme.
- Example 1 – infinite plate.
- Example 2 – infinite cylinder.
- Example 3 – stimulated vibrations of a string.
- Thermal problem for a ball with variable external temperature.
- Method of Finite Integral Transformations (FIT). (Scheme by G.A. Grinberg)
- Neumann’s problem for a rectangle.
- Bar vibration problem.
- A ball heating problem.
- Electrostatic problem for a sector of the circle. U (φ = 0) = Vr / a, 0 <φ <π / 2.
- Dirichlet’s problem for a sector of the circle. U (φ = 0, α.) = F (φ), 0 <φ <α.
- Dirichlet’s problem for a sector of the circle. General case.
- Problem solving, midterm exam.

**Special functions. Based on parts 3,4,5,6 (3 weeks)**- Bessel functions. Definitions, the radius of convergence for the series.
- Recurring relationships.
- Linear dependence (independence) of the Bessel functions with negative and positive index.
- Weber (Neumann) functions Y
_{ν}(z) - Asymptotic of Bessel functions 1st and 2nd kind for small and infinite argument.
- Integral representation.
- Bessel functions of semi-integer order.
- Bessel equation with parameter.
- Some useful integrals with Bessel function.
- SLP based on the Bessel equation.
- Fourier-Bessel and Dini series
- Dirichlet problem for a finite cylinder.
- Thermal flow problem for an infinite cylinder.
- Modified Bessel functions. Properties, recurring relationships, particular cases, asymptotic.
- Mc-Donald functions.
- Dirichlet problem for a short cylinder.
- Inhomogeneous Neumann problem for a finite cylinder.
- Legendre polynomials. Definition, properties.
- Legendre equation, show that Pn is a solution.
- General solution of Legendre equation. Second solution.
- SLP based on Legendre equation.
- Internal Dirichlet problem for a sphere.
- External Dirichlet problem for a sphere.
- A sphere within an ideal liquid flow.
- A point charge near the surface of a conducting sphere.
- Associated Legendre polynomials.
- SLP based on associated Legendre functions.
- Problem solving, midterm exam.

**Green’s function and its relationship with the FIT method. Based on part 5. (2 weeks)**- Generalized functions, Delta-Dirac function.
- Inhomogeneous differential equations. Green’s function method.
- Green’s function in quantum theory and perturbation theory.
- Solution in general case.
- Modified Green function.
- Type II Green function.
- Fundamental solution for the Laplace operator. Space dimension = 1,2,3.
- Relationship between Grinberg’s methods (FIT) and Green’s method.
- Problem solving, midterm exam.

**Bibliography:**

- G.B. Arfken, H.J. Weber, “Mathematical methods for physicists” Elsevier Academic Press 2005
- E. Butkov “Mathematical Physics” Adisson-Wesley, Readings, Massachusetts 1968
- P.Dennery, A. Krzywicki “Mathematics for Physicists” Harper & Row, New York 1967
- J.Mathews, R.L.Walker, “Mathematical Methods of Physics” Carnegie-Melon U 1984
- P.M.Morse, H.Feshbach, ,“Methods of Theoretical Physics” V.1,2, McGraw Hill, New York 1953
- A.J.Diaz, A.U.Araujo “Funciones Especiales” Universidad de Sonora, 2007
- Robert D. Richtmyer, “Principles of advanced mathematical physics”, Spring-Verlag.