Syllabus
1. Sequences and series of functions
Pointwise and uniform convergence; criteria for uniform convergence of sequences and series of functions; interchanging of limits, derivative and integral of sequences and series of functions; power series; real analytic functions. 2. Lebesgue integral
Sigma-algebras, measures; construction of the Lebesgue measure; measurable functions; approximation of measurable fuunctions by simple functions; integral of simple non-negative functions; integral of general functions and its properties; limite passage through the integral; relations among Riemann, Newton and Lebesgue integral; integral dependent on parameters; Fubini's theorem, change of variables. 3. Lebesgue spaces
Definition, norms, basic properties. Dense subsets. Mollifier. 3. Line integral in general dimension
The notion of a curve, line integrals of 1st and 2nd kind. Potential and curl-free vector fields. 4. Surface integral in general dimension
The notion of a surface, orientation of a surface. Surface integrals of 1st and 2nd kind, Gramm determinant, Gauss-Ostrogradskij, Green and Stokes theorems. Integral representations of div and curl operators. 5. Fourier series
Orthogonal polynoms. Abstract Fourier series. Trigonometric Fourier series.
Annotation
Basic mathematics course for 2nd year students of physics. Prerequisities: Mathematical analysis I+II, NOFY151,
NOFY152, and Linear algebra I+II, NOFY141, NOFY142.