LEARNING OUTCOMES

The course introduces general mathematical notions and tools at the basis of engineering modeling, related to the study of the functions of one or more real variables. In particular, the concept of limit and continuity, the differential and integral calculus, also of functions of several real variables, the resolution of ordinary differential equations, the analysis of curves and surfaces, and the study of the convergence of numerical series.

AIMS AND LEARNING OUTCOMES

The "Mathematical Analysis I" course aims at giving basic mathematical tools necessary to the studies in the engineering field.

At the end of the lessons the student will have acquired sufficient theoretical knowledge:

• to identify, understand, formulate and solve general engineering problems related to mathematically modeled quantitie using appropriate analytical methods;
• to combine notions of theory into practice to solve basic model engineering problems;
• to be able to autonomously learning new mathematical tools useful for engineering applications, throughout the entire working life;
• to analyze and model geometric and physical objects related to functions of one or more real variables, and to calculate quantities associated with them;
• to apply mathematical resolution tools in the context of the differential calculation of the functions of one or more real variables;
• to apply mathematical resolution tools in the context of the integral calculation of the functions of one real variable;
• to compute the maximum and minimum unconstrained values ​​of functions of one and several variables, useful in application areas of optimization;
• to analyze and model geometric objects related to curves, and calculate associated quantities;
• to understand and solve simple models related to ordinary differential equations, through which physical phenomena of engineering interest are represented;
• to know the concept of numerical series and to evaluate their convergence, useful in the approximate calculation of quantities in the numerical-computational field.

PREREQUISITES

Elementary algebra: literal calculus, polynomials, equations and inequalities, trigonometry.

TEACHING METHODS

72 hours of theoretical lessons, 48 hours of classroom practices. During the theoretical lessons the definitions and the theorems will be presented with many examples and applications. During the other part of the course many exercises will be solved. 

In addition, a tutor will solve some exercises in extra (optional) lesson hours.

SYLLABUS/CONTENT

The teaching program includes both theoretical study and practical resolution of exercises in the following topics:

• Sets, logic, real numbers, infimum and supremum
• Functions of one real variable, elementary functions, limits, infinitesimals and infinities, continuous functions, derivable functions, differentiable functions. Taylor’s formula, expansion of elementary functions.
• Primitives and indefinite integrals, methods of indefinite integration, definite integrals, fundamental theorem of integral calculus.
• Functions of several variables (scalar and vectorial fields), limits and continuity. Directional derivatives. Differentiable functions. Necessary and sufficient conditions for differentiability. Derivatives of composite functions. Derivatives of higher order, Schwarz Theorem and Taylor polynomial in several variables. Unconstrained maxima and minima of scalar fields, necessary and sufficient conditions, Hessian matrix.
• Differential equations of the first order, with separable variables, linear, homogeneous, Bernoulli and Riccati types. Existence and uniqueness theorem (hints) for the Cauchy’s problem. Linear differential equations. Linear differential equations of higher order with constant coefficients, homogeneous and non-homogeneous,
• Improper integrals of one variable.
• Numerical series. Convergence criteria for constant sign numerical series. Numerical alternating series and absolutely convergent series.

RECOMMENDED READING/BIBLIOGRAPHY

Handouts: "MATHEMATICS I"  and "MATHEMATICS II" by prof. Maurizio Romeo, downloadable for free from the web page of the course.

Sheets containing links to web pages with different solved exercises, downloadable for free from the web page of the course.

Workbook: Laura Recine - Maurizio Romeo, Esercizi di analisi matematica - Volume II, Maggioli Editore.

P. Marcellini – C. Sbordone: Calcolo, Liguori Editore, Napoli, or any other good text of mathematical analysis.

M.Baronti – F.De Mari – R.Van Der Putten – I.Venturi: Calculus Problems, Springer