28616 - Mathematical Analysis T-B (A-K)

Course contents

Course contents COMPLEX NUMBERS

Definition and operations on complex numbers.  Algebraic form. Trigonometric form . The exponential form of a complex number. Roots of a complex number, algebraic equations in C.

IMPROPER INTEGRALS AND NUMERICAL SERIES.

Definition of an improper integral, absolute convergence of an improper integral, comparison criterium. Real and complex numerical series, convergence and absolute convergence of a series, criteria of absolute convergence, Leibniz criterium, integral criterium.

FUNCTIONS OF SEVERAL REAL VARIABLES

Real and vector functions of several real variables: generalities. Definition of a continuous function and of the limit of a function. The Weierstrass theorem and the intermediate value theorem for functions of several variables. Partial derivatives. Differentiability and tangent plane. C^(1) functions. Jacobian matrix. The chain rule. Partial derivatives of higher order. The Hessian matrix. Taylor's formula of the second order for functions of several variables. Local extrema for real functions of several variables.

DIFFERENTIAL EQUATIONS.

Introduction to ordinary differential equations and the Cauchy problem. Linear differential equations: general integral for homogeneous and non homogeneous equations. Explicit solvability of first order equations and of second order equations with constant coefficients and Cauchy problems for first and second order linear equations. Equations with separable variables.

MULTIPLE INTEGRALS

Definition of the double integral for continuous functions on a compact rectangle. Properties of the double integral. Extensions to more general domains. Reduction theorems for double integrals on rectangular and normal domains. The theorem on the change of variables in a double integral. Triple integrals: (rough) extensions of definitions and theorems on double integrals to triple integrals.

Readings/Bibliography

Testi consigliati:
S. Abenda  - Analisi Matematica, Ed. Progetto Leonardo -Bologna
S. Abenda - Esercizi di Analisi Matematica Vol2, Ed. Progetto Leonardo -Bologna

 M. Bertsch, R. Dal Passo, L. Giacomelli - Analisi Matematica, ed. McGraw Hill.
Potrà essere utile integrare il testo con un libro di esercizi, a scelta dello studente.

V.Barutello, M.Conti, D.L.Ferrario, S.Terracini, G.Verzini- Analisi Matematica 2 ed. Apogeo.

Teaching methods

Lectures by the course instructor. Exercises solved and discussed by the instructor.

Assessment methods

The evaluation of the course is based on both a written and an oral examination.

Teaching tools

Textbook; exercises and online material  available at the address: 
http://www.dm.unibo.it/~baldi  and here.

Links to further information

http://www.dm.unibo.it/~baldi

Office hours

See the website of Annalisa Baldi