28616 - Mathematical Analysis T-B

Learning outcomes

Know 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

Complex numbers: Introduction, algebraic and trigonometric form, Gauss plane, De Moivre formula, n-th roots of a complex number. Differential calculation for functions in several variables: Introduction Elements of topology in R ^ n. Functions from R ^ n to R ^ m (n, m = 1,2,3). Limits and continuity. Bolzano theorem. Weierstrass theorem. Multi-variable functions with real values Partial and directional derivatives for multi-variable functions with real values. Gradient and its properties. Higher order derivatives. Hessian. Schwarz's lemma. Taylor's second-order formula. Tangent plane. Differential calculus for functions in several variables with vector values. Jacobian. Composition of functions: Jacobian theorem of the composite function. Differential calculus applications: Relative maxima and minima. Fermat's theorem. References on quadratic forms associated with symmetric matrices and their classification. Critical point classification: necessary or sufficient conditions for C ^ 2 functions. Measurement and integration for functions in several variables Peano-Jordan measurement. Riemann integral for functions from R ^ n to R. Properties of the integral: additivity, monotony, linearity. Average theorem. Double and triple integral reduction theorems in normal domains. Knights principle. Knights theorem. Variable change for multiple integral. Polar, spherical, cylindrical coordinates. Curves in parametric form and curvilinear integrals Regular curves. Regular curved lines. Orientation and oriented curves. Curved integrals on oriented curves: length, curvilinear integral of a function. Vector fields closed fields and exact fields, rotor and divergence of a vector field, work and potential of a vector field. Elements of surface integrals and Gauss-Green theorem: regular parametric surfaces, area and integral of a scalar function on a parametric surface, oriented surfaces, flow of a vector field through an oriented surface, Gauss-Green theorem.

Readings/Bibliography

M. Bramanti, C. Pagani, S. Salsa: Analisi Matematica 2 - Zanichelli 2009

Teaching methods

Lectures and frontal exercises

Assessment methods

Exam dates are published on Almaesami The exam consists of two parts. The first part consists of 4 multiple choice exercises and one with course, while the second part (which will take place a few days later) consists of 2 theoretical questions on course topics. The mark in each of these two parts is out of thirty and is considered passed with a mark greater than or equal to 18 for each of the two parts. The duration of the first part of Analysis is 2 hours. The duration of part II of Analysis is 1 hour. The final mark is obtained as an average of the two tests if they have all been passed. If the student scores more than 30 points, the final score 30/30 with honors is entered on Almaesami Registration on Almaesami is mandatory for both parties using the registration lists open on Almaesami. It is mandatory to present yourself at the exam with a photo ID. At the end of the correction of the written tests, a special student reception is set for viewing the tasks and, at the end of this reception, the Commission proceeds to record the valid marks. Further information on the exam procedures and on the scores of the individual exercises as well as the publication of the exam dates can be found on the teacher web pages. The IOL pages also publish some exam type texts relating to the part of Mathematical Analysis. -------------------------------------------------- ----- Therefore Part A (2 h). The student carries out exercises of the program carried out in the form of guided answer exercises and in full. During this part of the test, the student can only consult their own textbooks and Mathematical Analysis notes, while the use of any electronic device is prohibited. The student who reaches the admission threshold (i.e. 18/30) is admitted to part B. Part B (duration 1 hour). The student can bring only the pen with him, he answers in writing to four theoretical questions on topics developed during the course. The score for this part ranges from 0 to 30 + optional question points. Test passed with a mark greater than or equal to 18. During this test, the consultation of notes or other support is not allowed. Further information on the exam tests (including the scores of the individual exercises and the exam tests calendar) are available on the teaching website: http://www.unibo.it/docenti/cataldo.grammatico. Exam dates are published on Almaesami. The standard texts of some part A exams are available on IOL. Office hours See the Grammatico Cataldo website [https://www.unibo.it/sitoweb/cataldo.grammatico]

Teaching tools

The IOL pages also publish some exam type texts relating to the part of Mathematical Analysis.

Office hours

See the website of Cataldo Grammatico