27993 - Mathematical Analysis T-2

Learning outcomes

The student knows the basic concepts and the main properties of functions of several real variables, in particular differential and integral calculus.

Course contents

Essential prerequisites of the course are knowledge of all topics covered in the Mathematical Analysis T-1 course, as well as many topics covered in the Geometry and Algebra T course (vector spaces, linear transformations, matrices, determinants, analytic geometry in the plane and space).

1. Differential equations. Cauchy's problem for differential equations and systems. Theorems of existence, uniqueness and prolongability. Solution methods for nonlinear equations with separable variables and for first-order linear equations. Space of solutions of a homogeneous nonhomogeneous linear differential equation of order n. Solving nonhomogeneous second-order linear differential equations by similarity method.
2. The n-dimensional Euclidean space. The structure of vector space, scalar product and Euclidean norm. Elements of topology.
3. Limits, continuity and differential calculus for functions of several real variables. Real and vector functions of several real variables: generalities. Definition of limit and continuous function. Weierstrass theorems, intermediate values for functions of several variables. Definition of partial derivative and directional derivative. Differentiable functions and functions of class C^1; the differential and the Jacobian matrix. The theorem on the differentiability of a compound function. Higher-order partial derivatives. Second-order Taylor formula for functions of several variables. Relative extrema for free real functions of several real variables.
4. Multiple integrals. Definition of Riemann double integral on finite and measurable sets. Properties of the double integral. Reduction theorems on rectangles and on simple sets. The theorem of change of variables. Triple integrals: extension of definitions and theorems on double integrals. Notes on generalized double integrals.
5. Curvilinear and surface integrals. Regular and regular piecewise curves, length of a curve, integral of a function on a curve. The integral of a vector field on an oriented curve. Conservative vector fields and their potentials. Green-Gauss theorem. Regular and regular surfaces at strokes in R^3, area of a surface, integral of a function on a surface. Flow of a vector field through an oriented surface. Divergence and Stokes theorems. Curl fields and potential vector.

Readings/Bibliography

Teoria:

• M. Bertsch, A. Dall'Aglio, L. Giacomelli, Epsilon 2. Secondo corso di Analisi Matematica, ed. McGraw-Hill Education (2024).
• G. Barozzi, G. Dore, E. Obrecht, Elementi di Analisi Matematica Vol. 2, ed. Zanichelli (2015).
  

Esercizi:

• M. Bramanti, Esercitazioni di Analisi Matematica 2, ed. Esculapio (2012).

Teaching methods

Frontal lectures to explain the basic notions, examples, and counterexamples.

Exercises solved by the teacher.

Additional exercise sheets for personal study.

Office hours.

Assessment methods

The final exam for the course consists of a written test and an oral test, both of which are compulsory and to be taken in that order.

The written test aims to test the ability to apply theory to solving exercises of the type of those proposed during the course. Passages should be reported and justified. No books, notes or calculators are allowed; paper and pen only. It lasts 2 hours and 30 minutes.

The written test is passed with a grade greater than or equal to 18/30.

If the written test is passed, one can proceed to the oral test. This aims to test the knowledge and understanding of the theory developed during the course. You will be asked to give definitions and examples of concepts and to give statements and demonstrations of theorems.

The oral test is to be taken within the same roll call as the written test passed. Failure or absence in the oral will result in forfeiture of the validity of the score obtained in the written.

There will be 6 calls (each of which will have both written and oral parts): 4 in the summer session (June, July and September) and 2 in the winter session (January and February).

Teaching tools

Tutoring (TBA) and office hours.

Additional material will be made available on the Virtuale page of the class.

Office hours

See the website of Nicola Abatangelo

See the website of Eugenio Vecchi