Mathematical analysis II.1

mathematics
physics

obligatory courses

(in Polish) Oczekuje się dobrej znajomości zagadnień ujętych w sylabusach przedmiotów Analiza matematyczna I.2 oraz Wstęp do matematyki.

Many-variable differential calculus, measure and integration theory.

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.

M.Spivak, Modern Approach to Classical Theorems of Advanced Calculus

W.A. Benjamin, L.Bers, Calculus

W.Rudin, Principles of Mathematical Analysis, McGraw-Hill Science Engineering

W.Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. xi+412 pp.

1. A student has to be familiar with the notions of directional derivative, partial derivative, differential of a map and its Jacobi matrix. It is necessary to know what are the relations between the mentioned notions and to know their algebraic and analytic properties. A student must be able to show examples illustrating relations between partial derivatives and the differential of the map.

2. A student knows Schwarz theorem about symmetry of the second differential of the map, knows Taylor theorem, he or she is able to give a sufficient condition for existence of a locally extreme values of a real valuied function of severak veriables. A student knows how to find the least upper bound or the greates lower bound of a real valued function defined on variuos subsets of the euclidean space. A student is able to determine a type of o critical point of a C^2 function.

3. A student knows the Implicit Function Theorem, the Inverse Map Theorem, the notion of difefomorphism and the notion of a manifold embedded in the euclidean space.

4. A student is able to write down explicite formulas for diffeomorophism mapping one open subset of R^2 or R^3 onto another in simple cases. A student is able to prove that some sets defined by a few equations are manifolds and the other are not.

5. A students knows what are Lagrange multipliers and is able to use the Langrage method of finding constrained local extrema of a real function of several variables.

6. A student knows basic notions of Lebesgue measure theory including Monotone Lebesgue Convergence Theorem, Dominated Lebesgue Convergence Theorem, Fubini's Theorem, change of variables formula. He or she is able to use the above theorems for investagating continuity or smoothness of parameter dependet integrals and is able to give examples showing that hypothesis of the theorems cannot be dropped .

On the basis of scores obtained during the semester and the final exam.