ANALISI MATEMATICA II

The goal of the course is to give students elements and fundamental techniques useful to study sequences and series of functions, calculate limits of functions of several variables, find maximum and minimum of functions, solve some kinds of differential equations, calculate integrals of functions of two or three variables, study the differential forms, calculate the integral of a differential forms,  in accordance of the goal 4 of the Agenda 2030. 

In particular, the learning objectives of the course are:

1. Knowledge and understanding: The student will learn some concepts of Mathematical

Analysis and will develop both computing ability and the capacity of manipulating some

mathematical structures, as limits, derivatives and integrals for real functions of

more real variables.

2. Applying knowledge and understanding: The student will be able to apply the acquired

knowledge in the basic processes of mathematical modeling of classical problems arising from

Engineering.

3. Making judgements: The student will be stimulated to autonomously deepen his/her knowledge

and to carry out exercises on the topics covered by the course. Constructive discussion between

students and constant discussion with the teacher will be strongly recommended so that the

student will be able to critically monitor his/her own learning process.

4. Communication skills: The frequency of the lessons and the reading of the recommended books

will help the student to be familiar with the rigor of the mathematical language. Through constant

interaction with the teacher, the student will learn to communicate the acquired knowledge with

rigor and clarity, both in oral and written form. At the end of the course the student will have

learned that mathematical language is useful for communicating clearly in the scientific field.

5. Learning skills: The student will be guided in the process of perfecting his/her study method.

 

In particular, through suitable guided exercises, he/she will be able to independently tackle new

topics, recognizing the necessary prerequisites to understand them.

Blackboard lessons with related sessions of exercises.

1. (1 cfu). Sequences and series of functions. Pointwise and uniform convergence of sequences. Theorems of continuity, derivability and integration. Series of functions. Pointwise, absolute, uniform and total convergence for series of functions. Power series. Cauchy - Hadamard and D'Alembert theorem*. Theorem of Abel*. Series of Taylor. Sufficient conditions for series of Taylor. Fourier series.
   

2. (2 cfu). Functions of n variable. Euclidean spaces. Functions in euclidean spaces. Limits of functions. Theorems regarding limits*. Continuity. Continuous functions and connect subsets.Theorem of existence of the zeros. Theorem of Weierstrass. Partial and directional derivatives. Differential functions. Differentiability and continuity. Theorem of the total differential*.  Derivatives of upper order. Theorems of derivability of composite function. Theorem of Schwarz*. Taylor's formula. Functions with zero gradient. Local extremal points. Necessary and sufficient conditions for local extremal points.
   

3. (1,5 cfu). Differential equations. Differential equations of order n. Cauchy problem. Solutions. Theorema of uniqueness of solution of Cauchy problem. Local and global existence and uniqueness for Cauchy problem*. First order equations: equations with separable variables, homogeneous equations, linear equations, Bernoulli equations. Linear differential equations of order n. Wronskiana matrix. Lagrange method.
   

4. (1cfu). Measure and integration. Riemann measure. Riemann integral. Integration of bounded functions. Formulas of reduction for multiple integrals. Change of variables*. Polar coordinates.
   

5. (0,5 cfu). Curves and differentail forms. Curves in R^2. Simple curves. Closed curves.  Regular curves. Rectifiable curves and length*. Curvilinear abscissa. Integral on a curve. Differential forms. Potential. Integral of a differential form. First integrability criterion*. Closed differential forms and exact differential forms. Second integrability criterion*. Gauss-Green formulas*. Divergence theorem.

1. Di Fazio G., Zamboni P., Analisi Matematica 2, Monduzzi Editoriale.

2. Fusco N., Marcellini P., Sbordone C., Analisi Matematica 2, Liguori Editore.

3. Bramanti M., Pagani C.D., Salsa S., Analisi Matematica 2, Zanichelli.

4. Fanciullo M. S., Giacobbe A., Raciti F., Esercizi di Analisi Matematica 2, Medical Books.

5. D'Apice C., Durante T., Manzo R., Verso l'esame di Matematica 2, Maggioli editore.

6. D'Apice C., Manzo R., Verso l'esame di Matematica 3, Maggioli editore.