MATHEMATICAL ANALYSIS 3
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Teaching methods
- Contacts/Info
Mathematical Analysis 1 and 2, Linear algebra and geometry, geometry 1.
The exam is divided into two parts Written exam: duration 3 hours with exercises (4/5) on the topics developed during the course in order to verify the level of skills acquired. Oral exam: after passing the written test to assess the level of knowledge reached.
The course is a natural continuation of the course in mathematical analysis 2. It aims to deepen the study and modern classical analysis begun in the previous year. The student will acquire a working knowledge of advanced analysis methods, the statements and major demonstrations, he will increase his skills and he will be able to solve exercises, even theoretical, related to the topics.
1) Sequence and series of functions. Uniform and total convergence. Theorem of double limit. Uniform convergence and differentiability. A nowhere differentiable continuous function. Weierstrass approximation theorem. Space of continuous functions on compact sets. Equicontinouos and equibounded sets. Ascoli-Arzelà theorem 2) Complements on ordinary differential equations, Peano existence theorem, Extension of solutions. Qualitative study of differential equation. Applications to geometric and physical problems. 3) Sigma-algebra and measure. Measurable functions. Integral of positive functions. Monotone convergence theorem. Fatou’s lemma. Integrable functions. Dominate convergence theorem. Lebesgue measure in R and R^n. Measure on algebra and semi-algebra. Caratheodory exstension theorem. Distribution function of measures in R. Product measure. Fubini and Tonelli theorems. Integral depending on a parameter. Outline of Gamma and Beta functions. Stirling formula. 4) Curves and surfaces. Length of a curve and surface area (formulas). Integration on curves. Differential forms. Integration of differential forms. Exact forms and closed forms. Necessary and sufficient conditions. Applications to differential equations. The Gauss-Green formula.
1) Sequence and series of functions. Uniform and total convergence. Theorem of double limit. Uniform convergence and differentiability. A nowhere differentiable continuous function. Weierstrass approximation theorem. Space of continuous functions on compact sets. Equicontinouos and equibounded sets. Ascoli-Arzelà theorem 2) Complements on ordinary differential equations, Peano existence theorem, Extension of solutions. Qualitative study of differential equation. Applications to geometric and physical problems. 3) Sigma-algebra and measure. Measurable functions. Integral of positive functions. Monotone convergence theorem. Fatou’s lemma. Integrable functions. Dominate convergence theorem. Lebesgue measure in R and R^n. Measure on algebra and semi-algebra. Caratheodory exstension theorem. Distribution function of measures in R. Product measure. Fubini and Tonelli theorems. Integral depending on a parameter. Outline of Gamma and Beta functions. Stirling formula. 4) Curves and surfaces. Length of a curve and surface area (formulas). Integration on curves. Differential forms. Integration of differential forms. Exact forms and closed forms. Necessary and sufficient conditions. Applications to differential equations. The Gauss-Green formula.
Classroom lessons (by projection or blackboard) and exercises to be carried out at home, which, in general, will be corrected by both the teacher and the exerciser.
The teacher receives the students for clarifications and insights by appointment to be fixed by writing to the institutional email address. emanuele.casini@uninsubria.it