D1: Knowledge and understanding.
The students will acquire the basic notions in mathematical analysis needed to deal with ordinary differential equations, multiple Riemann integrals, vector calculus, and differential forms. At the end of the course the students will know the fundamental results on the above topics.

D2: Applying knowledge and understanding.
The students will be able to solve simple exercises on this topic and also they will be able to produce elementary proofs of some simple properties in the topics of ordinary differential equations, multiple Riemann integrals, vector calculus, and differential forms.

D3: Making judgements.
At the end of the course, the student will be able to recognize and apply the most elementary techniques of differential equations, Riemann integral for functions of multiple variables, vector analysis, and differential forms. Additionally, they will be able to identify situations and problems in which such techniques can be advantageously utilized (simple models from physics and other disciplines).

D4: Communication skills.
The students will be able to express themselves in a correct way on elementary topics from ordinary differential equations, multiple Riemann integrals, vector calculus, and differential forms.

D5: Learning skills.
The students will be able to use the handbooks in this discipline.

Differential and integral calculus in one variable. Linear algebra. Metric spaces.

Ordinary differential equations. Riemann integral for functions of several variables. Curves and surfaces and their measures. Line and surface integrals. Vector analysis. Differential forms. Stokes, Gauss, and Gauss-Green formulas.

Main textbooks:

1) E. Giusti, Analisi matematica 2, Bollati-Boringhieri, 2003

2) C.D. Pagani, S. Salsa, Analisi matematica 2, Zanichelli, 2016

3) G. Catino e F. Punzo, Esercizi svolti di analisi matematica e geometria, volume 2, Esculapio,

4) N. Fusco, P. Marcellini e C. Sbordone, Lezioni di analisi matematica due, Zanichelli, 2020.

Ordinary differential equations. Theorems of existence, uniqueness, maximal solutions, exit from compact sets, global existence, continuous dependence on data. Differential equations with separable variables, Bernoulli equations, homogeneous equations. Systems of linear equations and higher order linear differential equations. Method of variation of constants. Applications: pendulum equation, predator-prey models, suspension bridges, motion of planets.

Riemann integral for functions of several variables. Characterization of integrable functions, reduction theorem. Measurable sets and integration over measurable sets, integration over normal domains. Change of variables theorem. Polar, cylindrical, spherical coordinates. Integration on unlimited domains. Applications to physics: center of gravity and moment of inertia.

Curves and surfaces, and their measure. Integration on curves and surfaces. Vector analysis. Differential forms. Integration of differential forms. Vector fields, curl, divergence. Conservative and solenoidal fields. Stokes, Gauss and Gauss-Green formulas.

Lectures. Classwork.

The final examination is designed to ascertain knowledge of the topics of the entire course programme. It consists of a written and an oral examination. The written examination consists of carrying out exercises on the topics covered during the course. The oral examination is aimed at ascertaining the student's theoretical preparation. In the written test, students will be asked to solve a number of exercises similar to those carried out in lectures and exercises, and assigned during the course. The mark for the written test will be expressed in thirtieths. The overall mark is the sum of the marks awarded for each individual exercise. The written test is conducted as follows: the test consists of solving exercises on the course topics. Notes, books, calculators or other calculating instruments, objects equipped with a camera or capable of connecting to a network are not permitted. The condition for admission to the oral test is to have obtained a mark of 15/30 or higher. Please note that the pass mark is 18/30. The mark obtained in the written examination remains valid for the session only (session means one of the three periods January-February, June-July, September) The written examination may be repeated: the submission of a written examination cancels any previous written examination. The oral examination will aim to test theoretical knowledge of the discipline, expression skills and language property. It will focus on the understanding of the definitions and statements of the theorems discussed in the lecture, and may include a discussion of the written paper and demonstrations of some theorems. A final grade will be awarded at the end of the oral examination, also taking into account the result of the written paper. The final grade is awarded by means of a mark expressed in thirtieths. The examination is passed with a mark of 18/30. In order to obtain the minimum mark (18/30), knowledge of the definitions and main results is required, together with the ability to solve elementary exercises with a reasonable command of the language of the subject. To achieve the maximum mark (30/30 with distinction), on the other hand, it is necessary to demonstrate excellent knowledge of all the topics covered in the course; to answer all the questions correctly; to carry out all the exercises correctly. For the use of examination aids by students with disabilities, specific learning disorders (DSA) special educational needs (BES), please contact the Disability Service or the University DSA Service in advance.