27993 - Mathematical Analysis T-2

Learning outcomes

At the end of the course, the student will have a good methodological and operating mastery of the institutional aspects of differential and integral calculus for functions of several variables.

Course contents

Differential equations

Solution methods for first-order linear differential equations and separable variables equations. The Cauchy problem for differential equations and systems: existence, uniqueness, and extendibility of solutions.

The Euclidean space Rⁿ

The structure of vector space, scalar product, and Euclidean norm. Open, closed, bounded, compact, and connected subsets of Rⁿ.

Limits and continuity for functions of several variables

Limits for sequences in Rⁿ. Real and vector functions of several real variables: generalities, limits, and continuity. The Weierstrass and intermediate values theorems.

Differential calculus for functions of several variables

Partial derivative and directional derivative. Differentiable functions and C¹ functions. Jacobian matrix. Differentiability of a composite function. Higher-order partial derivatives. Hessian matrix. Second-order Taylor formula for functions of several variables. Free and constrained local extremums.

Double and triple integrals

Measurable sets of the plane. Riemann integral of functions of two variables; reduction formulas, change of variables. Triple integrals: extension of the definitions and theorems on double integrals.

Curves and curvilinear integrals

Regular and piecewise regular curves. Length of a curve, curvilinear integral of a function. Oriented curves. Vector fields, conservative and irrotational vector fields. Work of a field. Gauss-Green and Stokes theorems in the plane.

Surfaces and surface integrals

Regular and piecewise regular surfaces in R³. Tangent plane and normal unit vector. Area of a surface, integral of a function on a surface. Oriented surfaces, flux of a vector field through a surface. Divergence and Stokes theorems.

Readings/Bibliography

Theory

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 2, Zanichelli.

M. Bertsch, A. Dall’Aglio, L. Giacomelli: Epsilon 2, Secondo corso di Analisi Matematica, Mc Graw Hill

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

Exercises

M. Bramanti: Esercitazioni di Analisi Matematica 1, Esculapio.

Teaching methods

Classroom lectures illustrate the fundamental concepts related to the course topics, complemented by examples and counterexamples to better understand the topics presented.

Solution of exercises in the classroom.

Assessment methods

The exam consists of a preliminary written test and an oral one.

The written test consists of 5 exercises related to the topics covered in the course. It lasts two and a half hours and it is passed by obtaining at least a score of 15 out of 30. To take the written test, students must register at least four days in advance via AlmaEsami.

The oral exam, following the written test, focuses primarily on the theoretical aspects of the course. Students must demonstrate knowledge of the concepts covered in the course (particularly definitions and theorems) and the ability to connect them.

There are 6 exam sessions (each with both a written and oral exam): 3 in the summer period (June and July), 1 in the fall period (September), and 2 in the winter period (January and February).

The written exam is valid for taking the oral exam only in the same period.

Students with specific learning disorders (SLD) or temporary/permanent disabilities

We recommend contacting the University Office responsible for support services in a timely manner (https://site.unibo.it/studenti-con-disabilita-e-dsa/en). The office will evaluate the students' needs and, where appropriate, propose possible accommodations. These must in any case be submitted for approval at least 15 days in advance to the course instructor, who will assess their suitability also in relation to the learning objectives of the course.

Teaching tools

Teaching materials made available on Virtuale.

Office hours

See the website of Giovanni Dore

See the website of Andrea Bonfiglioli