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. 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. Integration of differential forms Outer algebras on linear vector space, differential forms, differentiation, outer differential, integral from a differential form. General Stokes theorem.
Annotation
Basic mathematics course for 2nd year students of physics. Prerequisities: Mathematical analysis I+II, NMAF051,
NMAF052, and Linear algebra I+II, NMAF027, NMAF028.