Mathematical analysis II.2

mathematics
physics

obligatory courses

(in Polish) Oczekuje się dobrej znajomości zagadnień ujętych w sylabusie przedmiotu Analiza matematyczna II.1.

Change of variables in Lebesgue integral - multidimensional case. Integrals dependent on parameters, their differentiability with respect to parameters. Convolution. Weierstrass Approximation Theorem (e.g. Tonelli polynomials). Curves and surfaces in R^3: curvature and torsion, inner product. Lebesgue-Riemann measure on manifolds embedded in R^n, an example of a polyhedron with small edges and huge area insrcibed in a cylinder.

Examples. Mass center and Guldin Theorems. Vector analysis in R^3. Green's Theorem, Classical Stokes Theorem and Divergence (Gauss-Ostrogradski)

Theorem with simple physical applications, physical meaning of divergence and rotation. Path integrals independent of the paths. Orientable and

nonorientable manifolds in R^n. Remarks on differential forms and the general Stokes Theorem on manifolds with boundary.

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 able to integrate function of two and three variables using theorems that allows switch the order of integration and change of variables.

2. Students has to be familiar with the definition of measure on smooth manifold and its properties. A student is able to calculate area of a two-variable function graph and area of a parametric surface.

3. A student is familiar with differential forms and is able to manipulate differential forms. A students knows Stokes' theorem and its particular formulations: Green's theorem, Gauss-Ostrogradsky's divergence theorem and examples of applications of those theorems. A student is able to integrate differential forms on manifolds of R^n. A student uses Green's and Gauss-Ostrogradsky's formulas to solve various problems.

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