COURSE AIMS AND OBJECTIVES: The aim of the course is to introduce Riemann multiple integrals, as well as line and surface integrals. Special attention is given to double integrals and the main results about them: Lebesgue's characterization of Riemann integrability, Fubini's theorem and the Change of variables theorem. After that the course covers integration of vector fields and differential forms, and the classical theorems: Green's theorem, Gauss - Ostrogradski divergence theorem and Stokes theorem.
COURSE DESCRIPTION AND SYLLABUS:
1. Multiple integrals. Riemann integral of a bounded function on a rectangle. Basic properties. Integrability of continuous functions. Area of sets in R2. Sets of area zero and sets of measure zero. Lebesgue's characterization of Riemann integrability. Integration over bounded subsets of R2. Fubini's theorem. Functions defined by an integral. Change of variables in double integrals. Polar, cylindrical and spherical coordinates. Multiple integrals. (6 weeks)
2. Integration over curves and surfaces. Piecewise smooth paths and curves and their length. Integral of a real function (scalar field) along paths and curves. Integral of a vector field (differential 1-form) along a path. Independence of path of integration. Potential of a vector field (exactness of a 1-form). Angle form and winding number. Green's theorem. Integrating over surfaces in R3. Flux and divergence of a vector field. The divergence theorem of Gauss - Ostrogradski. Rotation of a vector field. Exterior derivative of a differential 1-form (a differential 2-form). Stokes's theorem. (7 weeks)
Basic notions are introduced and explained on examples at lectures. Specific problems and techniques for their solving are explained at tutorials.