MATHEMATICAL ANALYSIS 3
Mathematical Analysis 1 and 2, Linear algebra and geometry, geometry 1.
The exam consists of two parts: a three-hour long written test with 4/5 exercises on the topics discussed in the course in order to verify the level of skills acquired. An Oral exam to assess the level of knowledge reached. Only students who pass the written test can access the oral exam.
Students who reach a grade of 16/30 or higher are admitted to the oral exam.
During the semester a partial test will be held. The students who reach a grade of 16/30 or higher in the partial test will be exempted from having to solve the corresponding part of the final written test. The admission grade for the oral exam will be the weighted average of the partial test and of the written test. This partial exemption is only limited to the first two exam sessions.
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) Sequences and series of functions. Uniform and total convergence. The double limit theorem. Uniform convergence and differentiability. Power series. The Taylor series. Real analytic functions. The Ascoli-Arzelà theorem.
2) Ordinary differential equations: the local existence-uniqueness theorem for an initial value problem. Peano’s existence theorem, Extension of solutions. Sufficient conditions for global existence of solutions. Equations of order n. Linear equations of order n. Linear independence and the space of solutions.Qualitative study of differential equation.
3) Sigma-algebra and measure. Measurable functions. Integral of positive functions. Monotone convergence theorem. Fatou’s lemma. Integrable functions. Dominated convergence theorem. Lebesgue measure in R and R^n. Product measure. Fubini’s and Tonelli’s theorems. Integral depending on a parameter.
4) Surfaces and surface integrals. Flux. The Green-Gauss formula. The divergence theorem. Stokes’ theorem.
Frontal lectures to expose the theoretical contents; exercise sheets to be carried out at home, which will then be solved by both the teacher and the tutor.
The teacher receives the students for clarifications and insights by appointment to be fixed by writing to the email address marco.magliaro@uninsubria.it