Mathematical analysis II.2 1000-114bAM4a
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.
Main fields of studies for MISMaP
mathematics
Type of course
Prerequisites (description)
Course coordinators
Term 2023L: | Term 2024L: |
Learning outcomes
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.
Assessment criteria
On the basis of scores obtained during the semester and the final exam.
Bibliography
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.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: