27993 - Mathematical Analysis T-2

Academic Year 2016/2017

  • Moduli: Simonetta Abenda (Modulo 1) Cataldo Grammatico (Modulo 2)
  • Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Computer Engineering (cod. 0926)

Learning outcomes

At the end of the course, the student will possess basic knowledge in the differential and integral calculus for functions in more than one variable and their applications.

Course contents

Differential Calculus in more than one variable Introduction to topology, metric spaces, Banach spaces. Functions from R^n to R^m (n,m=1,2,3). Limits and continuity. Bolzano and Weierstrass theorems. Functions from R^n to R:partial derivatives, directional derivatives,properties of the gradient function, higher order derivatives, Hessian, Schwarz lemma, Taylor formula at second order, tangent plane. Functions with vector values: the Jacobian, the Jacobian of composite functions.

Applications of differential calculus Local minima and maxima. Fermat theorem. Definition and classification of quadratic forms associated to symmetric matrices, Sylvester theorem, classification of critical points: necessary/sufficient conditions for C^2 functions. Regular varieties in implicit form. Normal and tangent spaces. Dini theorem and the local parametrization of a variety. Conditioned extrema. Fermat theorem in such case. Lagrange multiplier theorem.

Measure and integration Peano-Jordan measure. Riemann integration for function from R^n to R. Properties of integration: additivity, linearity, monotonicity. The integral average theorem. Reduction theorems for double and triple integrals in normal domains. Cavalieri principle and Cavalieri theorem. The change of variable in the integral. Polar, spherical and cylindrical coordinates.

Series : General notions on numerical sequences and series: definitions, convergence criteria. General notions on sequences and series of functions: pointwise, absolute, uniform convergence, Weierstrass criterium. Power series in R e in C:m Abel lemma, radius of convergence, properties of the sum of a power series, real and complex analytic functions. Fouries series: approximations of periodic functions with series of trigonometric functions.

Curves in parametric form and curvilinear integrals Regular curves. Piecewise regular curves. Orientation. Curvilinear integrall on non oriented curves: length, curvilinear integral of a function (mass, barycenter, inertia moments). Vector fields and differential forms. Curvilinear integral of a differential form and work. Exact differential forms and conservative vector fields. Closed differential forms and irrotational vector fields. Potential of a conservative vector field. Poincarè lemma.

Readings/Bibliography

Simonetta Abenda, Analisi Matematica, Ed. Esculapio (Bologna)

Simonetta Abenda: Esercizi di Analisi Matematica, Ed. Esculapio (Bologna)

Teaching methods

Lessons and exercises at the blackboard

Assessment methods

The examination is written and consists of two parts. It is obligatory to enrol in the Amaesami list of exams for both parts of the exam.

The first part lasts for 2 and 1/2 hours and it consists of multiple choice and traditional exercises. Students may use their own books and notes. It is forbidden to use any electronic device. The highest rank of this part is 16. If the student achieves 6.1/16 or more in the A part, he/she may partecipate in the B part.

The B part lasts for 1 h and the student may take only the pen with him/herself. In this part, the student must solve one of the exercises of his/her A part and must answer to two theoretical questions following the proposed draft. The highest rank for this part of the exam is 21.

The final mark is obtained adding the part A and part B marks. Final marks greater than 30 will correspond to 30/30 cum laude on Almaesami.

The student may check his work during a special office hours before the verbalization of all valid marks.

Further piece of information on exams is available in the web pages http://www.unibo.it/docenti/simonetta.abenda .

The dates of the exams are published on Almaesami.

Facsimiles of the part A written examination are avalaible in the Alma Campus collection.

Office hours

See the website of Simonetta Abenda

See the website of Cataldo Grammatico