Syllabus
1. Ordinary differential equations Solution of an ODE; Cauchy problem for the ODE's; basic existence and uniqueness theorems; scalar equations of the first order - basic methods of finding solutions; linear equations of the nth order - fundamental system, variation of the constant, special right-hand side. Connection to the system of ODEs. Wronskian, Bernoulli and Euler equations.
2. Number series and power series Convergent/oscilatory/divergent number series; convergence criteria for series with non-negative terms and general terms; absolute and relative convergence; product of series. Elementary power series, derivatives and primitives to series. Taylor series. Solution of ODEs by Taylor series.
3. Functions of more than one variable Metric, norm, open and closed sets, closure, interior, boundary. Convergence, completeness, compactness, separability. Banach and Hilbert spaces. Continuity and uniform continuity, Heine theorem. Continuous functions on a compact set. Contractive mapping. Banach fixed point theorem. Theorem on the solvability of ODE. Limit and continuity. Partial and directional derivatives, total differential. Grad, div and curl. Exact differential equations, integration factor. Chain rule, change of variables. Mean value theorem, Taylor series. Local and global extrema, Lagrange multipliers. Implicit functions. Regular mapping.
4. Calculus of variations Functional, Gateaux derivative, Frechet differential. Euler-Lagrange equations. Necessary and sufficient conditions for minima of functionals. Convexity. Legendre transform. Hamilton equations.
Annotation
Second part of the basic course of mathematics for the students of general physics (bachelor study). Follows the course
NOFY151.