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Academic year
Didactic period
Secondo Semestre

Training objectives

Aim of the course is to provide the basic tools of mathematical analysis, especially for what concerns the integral calculus for functions of one or more real variables and its applications to the resolution of problems based on mathematical models.
At the end of the unit, students will know the theoretical contents and the methods proper of mathematical analysis calculus.
They should be able to properly apply the concepts learnt to the resolution of different kinds of problems and to identify the most appropriate approach to do so.
Students will have to master the mathematical language and the logical-deductive approach, demonstrating the ability to illustrate their problem solving strategies in a logical, effective, pertinent and synthetic way.


Full comprehension of all contents of the Analisi Matematica 1.A course is required.

Course programme

1) Algebra of complex numbers.
2) Geometry of complex numbers.
3) Complex exponential.
4) Roots and logarithms in the complex field.
5) Factorization of polynomials in the complex fields.
6) Limits and derivatives of complex valued functions.
7) Definition of Riemann's integral, subdivisions, lower and upper sums.
8) Integrable functions. Geometric meaning of the integral.
9) Integrability criteria. Integrability of monotone functions and continuous functions.
10) Monotonicity, additivity and linearity of Riemann's integral.
11) Integral mean value. Mean value theorem for continuous functions. Integral functions.
12) Primitives and indefinite integrals.
13) The fundamental theorem of calculus.
14) Integration by direct recognition.
15) Integration by parts.
16) Integration by direct and reverse substitution.
17) Primitive of elementary rational functions.
18) Integration of Rational Functions.
19) Some useful substitutions in the calculation of primitives.
20) What is an ordinary differential equation and what is a solution.
21) Lipschitz functions. Uniqueness of local solutions.
22) Differential equations with separable variables. First order homogeneous linear differential equations.
23) First order inhomogeneous linear differential equations.
24) Second order homogeneous linear differential equations with constant coefficients.
25) Second order inhomogeneous linear differential equations with constant coefficients.
26) Ad hoc methods for inhomogeneous linear equations.
27) Some examples of classes of integrable equations.
28) Functions of multiple variables, examples.
29) Elementry Vector calculus.
30) Calculation of limits for function of multiple variables.
31) Partial Derivatives.
32) First order approximation and differentiation.
33) Tangent vectors and tangent plane to the graph of a function of multiple variables.
34) Total differential theorem.
35) Gradient, divergence, rotor. Jacobian matrix.
36) Derivative of composite functions of multiple variables.
37) Riemann's integral in R^n.
38) Peano-Jordan measure.
39) Riemann's integral on Peano-Jordan measurable sets.
40) Multiple integrals as iterate integrals.
41) Integration by lines, by layers.
42) Change of variable in multiple integrals.
43) Polar, cylindrical, spherical coordinates.
44) Solids of revolution and Guldino Theorem for Volumes.
45) Calculation of volumes, center of gravity, moments of solids.
46) Generalized integrals.
47) Convergence criteria for generalized integrals.
48) Summary exercises.

Didactic methods

The course provides 48 hours of classroom lectures with presentation of theoretical aspects, applications and exercises on the blackboard.
There will also be periodic tutoring sessions (two hours per week) with exercises and review of the topics.

Learning assessment procedures

The course learning assessment is done through a written test and an oral interview.

- In the written test the student is required to solve problems and exercises related to the course topics. The expected time for the written test is approximately 3 hours. It is not permitted to consult texts, use PC, tablet or smartphone. However, the student may still consult a personal hand written sheet (A4) in which he or she can write anything, that sheet must be delivered together with the test solutions. The student can use a pocket scientific calculator as long as it does not have graphic capabilities and can not be programmed. At the end of the written test, a score of at most 30 points is awarded. To pass the test and gain access to the oral interview you must get a score of at least 15 points.

- In the oral interview the student will be required to present some aspects of the course topics, illustrating some definitions, examples, properties, formulas, theorems, proofs, or applications. More than the mnemonic knowledge of the topics, we want to evaluate the logical understanding of concepts, the accuracy and rigor of the mathematical language used to describe them, and the ability to grasp the relationship between abstract aspects and concrete applications. The time for the oral test is about 30 minutes. If the outcome of the interview is not considered sufficient, the student will be able to retry it at a later date without necessarily having to repeat the written test.

The final vote is proposed at the end of the oral interview and will take into account all the elements that allow the teacher to evaluate the student's preparation: active participation during lectures, correctness and completeness in the written test, quality of exposure in the oral interview.

Passing the exam certifies to have acquired the knowledge and skills specified in the learning objectives of the course.

Reference texts

M.Bertsch, R.Dal Passo, L.Giacomelli - "Analisi Matematica" -McGraw-Hill (ISBN978-88-386-6234-8).
M.Bramanti, C.D.Pagani,S.Salsa-"Analisi matematica 1" con elementi di geometria e algebra lineare - Zanichelli (ISBN 978-88-08-25421-4)
Exercises: Marco Bramanti Esercitazioni di Analisi Matematica I Società Editrice ESCULAPIO ISBN 978-88-7488-444-5