31586 - Integrational Elements of Mathematical Analysis and Elements of Probability Calculation T

Learning outcomes

At the end of the course, after having passed the final assessment test, the student possesses the basic knowledge relating to the calculation of real multi-variable functions (properties, maxima and minima), curves, potentials, multiple integrals, their meaning, solution of some simple types of differential equations. Furthermore, he/she has the elementary notions of probability, with particular reference to some continuous distributions (uniform and normal distributions).

Course contents

Complex numbers: Introduction, algebraic and trigonometric form, Gauss plane, De Moivre formula, n-th roots of a complex number. Differential calculus 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. Real-valued multivariable functions Partial derivatives and directional derivatives for real-valued multivariable functions. Gradient and its properties. Higher order derivatives. Hessian. Schwarz's lemma. Second order Taylor formula. Tangent plane. Differential calculus for vector-valued multivariable functions. Jacobian. Composition of functions: Jacobian theorem of the composite function. Applications of Differential Calculus: Free relative maxima and minima. Fermat's theorem. Review of quadratic forms associated with symmetric matrices and their classification. Classification of critical points: necessary or sufficient conditions for C^2 functions. Constrained maxima and minima: Lagrangian function, necessary conditions for a point to be a constrained extrema with one or two dimensional constraint in space Measurement and integration for multivariable functions Peano-Jordan measurement. Riemann integral for functions from R^n in R. Properties of the integral: additivity, monotonicity, linearity. Mean theorem. Reduction theorems of double and triple integrals in normal domains. Cavalieri principle. Cavalieri's theorem. Change of variable for the multiple integral. Polar, spherical, cylindrical coordinates. Linear ordinary differential equations of the 1st and 2nd order 1st order equation: associated homogeneous equation, general integral and solution formula, Cauchy problem for a 1st order equation. second order equation with constant coefficients: associated homogeneous equation, general integral of the associated homogeneous equation, existence and uniqueness of the Cauchy problem, particular solution by Lagrange method and by sympathy, Wronskian and its properties.

Readings/Bibliography

Robert A. Adams: Calcolo differenziale 2 - Casa Editrice Ambrosiana distribuzione Zanichelli 2014

Teaching methods

Lectures and frontal exercises

Assessment methods

The exam takes place in written form and consists of two parts to be taken in the same session. In the first part the student solves multiple choice and/or open-ended exercises while the function study is carried out. In the second part, complete one of the questions corresponding to a theory topic from a list of questions published on virtual. The use of any electronic device connected to the internet during the exam is prohibited, under penalty of cancellation of the exam itself. The final score is the sum of the scores obtained in the two parts and is published on Almaesami. Students can appear for all exams. The exam dates are published on Almaesami. Registration on Almaesami for both parts of the exam is mandatory. Learning assessment methods Detailed information on the exam method Part A (duration 30 minutes). The student can only bring the pen with him and explains two theoretical topics following the path assigned by the teacher. The maximum score for this part is 5. Part B (duration 2 hours): Consists of multiple choice and/or open-ended or follow-up exercises. During part A, the student cannot consult his textbooks and notes and cannot use any type of calculator. The use of any other electronic device is prohibited. The maximum score for this test is 26, while the theory question has a score from 0 to 5. Multiple choice exercises are worth: +4 (correct answer), 0 (answer not given) -1 (wrong answer) Exercises with execution: calculation of an integral with a score from 0 to 5, bound extremants with a score from 0 to 5. Marking and verbalisation: The final mark is given by the sum of the scores obtained in the two tests. Scores above 30/30 will be recorded as 30/30 cum laude on Almaesami. At the end of the marking of the written tests, a special student reception is set up to view the homework and, at the end of this reception, the Commission proceeds to record all the valid marks. To refuse the grade it is necessary to participate in the homework viewing and communicate it verbally on the same day as the homework viewing. Exam timetable: it is published on Almaesami and visible on the Course of Study web page dedicated to exam sessions. The standard texts of some part A exam tests are published in the virtual spaces of the course. Reception hours Consult the Grammatico Cataldo website [https://www.unibo.it/sitoweb/cataldo.grammatico]

Teaching tools

Some standard exam texts relating to the Mathematical Analysis part are also published on virtual.

Office hours

See the website of Cataldo Grammatico

See the website of Giovanna Citti