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## Subject: MATHEMATICAL ANALYSIS C (A.A. 2021/2022)

### degree course in MATHEMATICS

Course year 2 9 Unit Analisi matematica C Theoretical Studies (lesson) TAF: Compulsory subjects, characteristic of the class SSD: MAT/05 CFU: 9 Teachers: Massimo VILLARINI oral final vote Italiano
Contents download ### Overview

The course presents the basic results on sequences and series of functions, implicit functions and ordinary differential equations. These topics are the basic knowledge requested to a student to progress in mathematics, and moreover will make him able to apply such skills to build, implement and study theoretically many mathematical models of frequent use in scientific and technological settings.

Sequences and series of real numbers. Single and multivarable differential and integral calculus. A working knowledge of basics of mathematical analysis concerning calculus of functions of one and several variables and of linear algebra, as taught at the first year in a university course majoring in mathematics

### Course contents

Sequences and series of functions
Sequences of functions: uniform convergence and Ascoli-Arzelà’s theorem. Applications to Peano’s existence theorem for ode’s.
Series of functions: pointwise, uniform and total convergence, term by term differentiation and integration.
Power series: radius of convergence, pointwise and uniform convergence, differentiation and integration of power series. Taylor series.
Fourier series: Fourier's coefficients (definition, heuristics, geometric interpretation), Riemann-Lebesgue Lemma, Dini's Theorem on pointwise convergence through Dirichlet's kernel.

Ordinary differential equations: existence (Picard's Theorem), uniqueness and continuation (Gronwall's Lemma). Contnuous and differentiable dependence on data.

The Implicit Function Theorem
Dini’s theorem in codimension one, general version for functions from Rᴺ+ᴹ to Rᴺ. The inverse function theorem; Lagrange’s multipliers for constrained maxima and minima.

### Teaching methods

Classroom lectures and exercise sessions. For any conceptually homogeneous set of topics of the course there will be homeworks that students could do individually or in collaboration: they will not enter in the evaluation process, but they will help students to understand their learning process. Corretions will be provided by the theacer both as individual or common sessions, at student's choice.

### Assessment methods

Written and oral exam. The written exam will be the first part of the whole, and only if successfull, up to theacer's evaluation, will give student's access to oral examination. Written examination will consists of exercises splitted in several parts: the first part will concern the very basic knowledge of the topics of the course, and their straightforward applications. The other parts of the written examination will try to see how deep is student's understanding of the course matter. The oral examination will focus on the theoretical part of the course, cheking student's skill in understanding and exposing theorems and results, as well as with its ability to apply them to simple example and exercises. Once the student has reached a sufficiently successful result in written exam, the contribution of each part of the exam to its evaluation is not fixed in advance and is left to theacer's judgement.

### Learning outcomes

Knowledge and understanding: Students will acquire theoretical and working knowledge on sequences and series of functions, ordinary differential equations and implicit functions.
Applying knowledge and understanding: Students will be able to model and solve problems related to the field of study.
Making judgments: Students will be able to evaluate coherence and correctness of results and recognize resolution methods appropriate to different types of problems related to the field of study.
Communication skills: Students will be able to communicate in a clear, precise and complete way mathematical statements in the field of study using approprtiate mathematical language and formalism.
Learning skills: The student will develop skills of independent learning and study of side arguments to those presented in the course.