Mathematical analysis II.1* 1000-113bAM3*

Linear and topological structure of Euclidean spaces; transformations, continuity. Calculus in several variables: directional derivative, differentiability, higher-order derivatives, symmetry od the second and higher order differentials, Taylor's formula, the implicit function theorem, local extrema. Manifolds in R^n, tangent spaces, local parametrizations and maps, manifolds defined by a system of equations, normal vectors. Constrained maxima and minima, Lagrange multipliers with
examples.The concept of measure; outer measure and Caratheodory's theorem. Lebesgue measure; measurable functions, Lebesgue integral. Lebesgue monotone
convergence theorem, Lebesgue bounded convergence theorem, the Fatou lemma. Fubini's theorem, change of variables under the integral.