28616 - Mathematical Analysis T-B

Academic Year 2022/2023

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Mechanical Engineering (cod. 0927)

Learning outcomes

The Student knows the methodological-operational aspects of mathematical analysis, with particular regard to the functions of several real variables and to differential equations, in order to be able to use this knowledge to interpret and describe engineering problems.

Course contents

Improper integrals for functions of one real variable.

Definition of improper integral.
Existence criterion; comparison criterion; absolute convergence criterion.


Numerical series.
Definition of numerical series; geometric series. Necessary conditions for convergence.
Existence criterion; integral criterion; comparison criterion; absolute convergence criterion; Leibnitz criterion; ratio criterion.

Limits and continuity for functions of several real variables with real and vector values.
R^n topology.  Real and vector valued functions of several real variables: generalities, limits and continuity.
Weierstrass and intermediate values theorems.

Differential calculus for functions of several real variables with real and vector values.
Directional derivatives, partial derivatives and differentiability for functions of several variables; Jacobian matrix, gradient. 
Derivation rules. Partial derivatives of higher order. Schwarz's theorem, Hessian matrix. Taylor's formulas up to the second order for functions of several variables.
Free global and local extremants for real functions of several variables: definitions, necessary conditions, sufficient conditions. Nature of the critical points.
Constrained global and local extremants for real functions of several variables: definitions and methods; manifolds in R^n, Lagrange's multipliers theorem.

Integral calculus for functions of several real variables.
Multiple integrals on rectangles; Reduction Theorem.
Multiple Riemann integral in bounded regions of R^2 and R^3; properties of the integral.
Reduction theorem on simple regions. Diffeomorphisms, Change of variable Theorem for multiple integrals. Polar coordinates; cylindrical coordinates; spherical coordinates.

Complex numbers.
The field of complex numbers; algebraic and trigonometric form of a complex number. Geometric interpretation of sum and product of complex numbers. Complex exponential. N-th roots of complex numbers. Equations in C.

Ordinary differential equations.
First order linear differential equations, explicit solution formula.
Homogeneous and non-homogeneous linear differential equations of higher order; general solutions and particular solutions; Existence and uniqueness theorems for the Cauchy problem; constant coefficients equations, resolutive methods.
Equations with separable variables.

Readings/Bibliography

Barozzi, Dore, Obrecht, Elementi di Analisi Matematica vol. 1 e 2, Zanichelli, Bologna.

M. Bramanti - C.D. Pagani - S. Salsa, Analisi Matematica vol. 1 e 2, Zanichelli, Bologna.

Salsa - Squellati, Esercizi di Analisi Matematica vol. 1 e 2, Zanichelli, Bologna.

Bramanti, Esercitazioni di Analisi Matematica vol. 1 e 2, Esculapio.

Teaching methods

Frontal lectures and exercises.

Assessment methods

The examination consists of a written test consisting of exercises and theoretical questions related to the topics covered in the course. The student must demonstrate knowledge of the concepts explained in the course (in particular definitions and theorems) and know how to apply them to concrete cases. You need to go to the exam with a university card and an identification document. It is not permitted to keep books, notes, calculators, cell phones or other materials with you. To take the written test you must register on the list, in the indicated time window, via AlmaEsami [http://almaesami.unibo.it/]. For the examination test calendar, always refer to AlmaEsami.

 

 

Office hours

See the website of Francesco Uguzzoni