# Mathematics II

**Academic Year 2022/2023**- Teacher:

**Salvatore D'ASERO**

## Expected Learning Outcomes

**Knowledge and understanding:**

students will learn some basic mathematical concepts and will develop the skills of calculation and manipulation of the most common objects of Mathematical Analysis: among these, integrals for functions of one or several real variables, the differential equations and the differential calculus for real functions of two or several real variables.

**Applying knowledge and understanding:**

through examples related to applied sciences, the student will be able to appreciate the importance of Mathematical Analysis as an important modeling tool.

**Making judgments:**

students will be able to deal with some simple but significant methods of Mathematical Analysis with sufficient rigor to refine their logical skills. Many demonstrations will be presented in a schematic and intuitive way to engage students and encourage them to achieve the goal on their own.

**Communication skills:**

by studying Mathematical Analysis, students will learn to communicate with rigor and clarity both orally and in written form. They will learn that using a correct language is one of the most important means of clearly communicating any scientific topic, not only mathematics.

**Learning skills:**

students, especially the most willing, will be stimulated to deepen some topics, also through group work.

## Course Structure

Lectures complemented by exercises

If the teaching is given in a mixed or remote way, the necessary changes may be introduced with respect to what was previously stated, in order to respect the program envisaged and reported in the syllabus.

Exams may take place online, depending on circumstances.

## Required Prerequisites

## Attendance of Lessons

## Detailed Course Content

**Integral calculus for functions of one variable**

Indefinite integral - Integration methods: integration by decomposition and sum, integration of rational functions, integration by parts, integration by change of variable - Definition of integral according to Riemann and its properties - Some classes of integrable functions - Definite integrals - Hints of theory of the Peano-Jordan measure - Geometric meaning of the Riemann integral - Fundamental theorem of integral calculus - Hints on generalized and improper integrals and their properties.

**Differential calculus for functions of two or several variables**

Recall of topology in the plane and in the euclidean n-dimensional space: internal points, external points and boundary points, open and closed sets, accumulation points and isolated points, bounded sets, compact sets, convex sets, connected sets by arcs, domain - Functions of several variables: limits and continuity - Differential calculus for functions of several variables: partial and directional derivative - Differential and differentiable functions - Higher order derivatives and Schwarz lemma - Differential operators: gradient, divergence, rotor, Laplacian - Differentiation theorem of composition of functions - Lagrange's theorem in R2 and characterization of functions with zero gradient in a region - Free extrema of a function of two variables and relative theorems - Search for absolute extrema on a compact set

**Integral calculus for functions of two or several variables**

Integral calculus for functions of several variables: double and triple integrals according to Riemann - Change of variables - Reduction formulas: Fubini's theorem - Integrals dependent on a parameter: Leibinz rule.

**Ordinary differential equations**

General information on differential equations - The Cauchy problem - First order differential equations - First order differential equations with separable variables - Cauchy's theorem on the existence and uniqueness of the solution - Second order linear differential equations with constant coefficients - Applications to the study of some mathematical models.

**Notes on the geometry of curves and on linear differential forms**

Regular and generally regular curves - Rectifiable curves and their length - Curvilinear abscissa - Curvilinear integral of a function - Linear differential forms - Curvilinear integral of a linear differential form - Exact and closed differential forms - Integrating factor - Applications.

## Textbook Information

- C. Canuto, A. Tabacco – Mathematical Analysis I – Springer-Verlag Italia
- C. Canuto, A. Tabacco – Mathematical Analysis II – Springer-Verlag Italia

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Integral calculus for functions of one real variable | 1 |

2 | Differential calculus for functions of two or more variables | 2 |

3 | Integral calculus for functions of two or more variables | 2 |

4 | Ordinary differential equations | 2 |

5 | Curves and differential forms | 2 |

## Learning Assessment

### Learning Assessment Procedures

N.B .: The verification of learning can also be carried out electronically, if the conditions require it.

### Examples of frequently asked questions and / or exercises

Knowing how to give a definition: of limit of functions with several variables; theorems on limits of functions; continuous functions and their properties; partial and directional derivative, differentiable functions and their properties, differential forms.

Knowing how to state and, if done in class, prove theorems: Schwarz's theorem, Cauchy's theorem for differential equations.