This course aims to illustrate the fundamentals of differential and integral calculus for functions of several variables, of the theory of numerical and functions series, of ordinary differential equations, as well as to introduce students to modeling and solving simple problems of practical interest which exploit these mathematical tools.
D1 - Knowledge and understanding skills
At the end of the course, the student should know the fundamentals of differential and integral calculus for functions of several variables, of numerical and function series, of ordinary differential equations.
D2 - Applying knowledge and understanding
The student should be able to deal with and solve simple exercises, questions, problems, of theoretical and computational nature, related to the topics of the course.
D3 - Making judgements
The student should be able to describe, model and solve simple problems of interest for applications, by using the mathematical tools developed during the course.
D4 - Communication skills
The student should be able to describe mathematical topics with an adequate command of language, as well as to translate practical problems in mathematical terms.
D5 - Learning skills
The student should be able to read and understand books and articles, by using the mathematical tools learned in the course, and be able to learn more advanced ones.

Differential and integral calculus for functions of one real variable. Basics of linear algebra and analytic geometry.

Euclidean spaces: linear, metric, topological structures.
Differential calculus in R^N: directional and partial derivatives, differentiation, differentiation rules, Taylor formula and applications, implicit function theorem and applications.
Power series and Taylor series expansions.
Integral calculus in R^N: integral on N-rectangles and their properties, measure in R^N, integral on bounded sets, reduction formulas, change of variables, generalized integrals.
Curves and surfaces: curves and surfaces in parametric or implicit form, length and area, line and surface integrals of a scalar field.
Vector calculus: vector fields, line and surface integrals of a vector field, curl and divergence, conservative fields, curl theorem, divergence theorem.
Differential equations: differential equations and mathematical modeling, Cauchy problem for ordinary differential equations, resolution by integration versus qualitative study, equations and systems, linear differential equations.

C. D. Pagani, S. Salsa, Analisi matematica, vol. 2, Masson, Milano, 1991. V. Barutello, M. Conti, D. L. Ferrario, S. Terracini, G. Verzini, Analisi matematica, vol. 2, Apogeo, Milano, 2008.
E. Giusti, Analisi matematica 2, Bollati-Boringhieri, Torino, 2003.

Sequences and Series of Functions Sequences of functions: pointwise and uniform convergence. Characterization of uniform convergence and continuity of the uniform limit. Limit-passing theorems under the integral and derivative signs. Series of functions: Taylor and Fourier series. Pointwise and uniform convergence. Necessary condition for uniform convergence. Weierstrass M-test, Leibniz criterion. Limit-passing theorems for series. Power series: domain and radius of convergence. Properties of the sum function. Term-by-term integration and differentiation theorems. Taylor series expandability. Conditions on derivatives. Analytic functions. Taylor expansions of main elementary functions. Metric Structure of ℝⁿ Euclidean distance, metric spaces. Accumulation, interior, and boundary points. Open, closed, and bounded sets. Functions from ℝⁿ to ℝᵐ, scalar and vector fields, level sets. Parametric curves and surfaces. Continuity and limits. Connected sets, connection and zero theorems. Compactness, Heine–Borel theorem, Weierstrass theorem. Vector spaces: scalar product, Cauchy–Schwarz inequality, Euclidean norm. Linear maps and associated matrices. Riesz representation theorem. Differential Calculus in ℝⁿ Directional and partial derivatives, Jacobian matrix, differential, and tangent plane. Differentiability and its properties. Gradient: direction of maximum increase. C¹ functions, differentiation rules, composite function, mean value theorem. Zero gradient, Cᵏ functions, Schwarz’s theorem. Linear and quadratic forms, Hessian matrix, Young’s theorem. Second-order Taylor expansion, local and global extrema. Criteria for critical points: eigenvalues and Hessian. Constrained extrema, Lagrange multipliers. Curves and surfaces in ℝ² and ℝ³, implicit and parametric equations. Implicit function theorem and local parametrization. Differential Equations Mathematical models. ODEs and PDEs. First-order ODEs: vector field, solutions, Cauchy problem. Existence and uniqueness theorems (Peano, Cauchy–Lipschitz). Separable variables, linearization method. Homogeneous and non-homogeneous ODEs, variation of constants, Bernoulli equation. Second-order ODEs: existence theorems, Newton equations, energy method. Linear ODEs with constant coefficients, characteristic equation, resolvent kernel, similarity method. Introduction to systems of ODEs. Riemann Integral in ℝⁿ Riemann-integrable functions on N-rectangles. Fubini’s theorem, reduction formulas for double and triple integrals. Peano–Jordan measurable sets, null sets, integrability on bounded sets. Normal sets in ℝ² and ℝ³, reduction formulas. Change of variables: polar, elliptic, and spherical coordinates. Generalized integrals in dimension 2: motivations, locally integrable functions, independence from the invading sequence, functions of constant or variable sign.

Lectures and classroom exercises. Tutorials. The teaching materials, including exercises and problems, are made available to students via Moodle platform and MS-Teams

Further information and classroom notes will be available in the Microsoft Teams devoted to the course.

Each exam consists of a written test and the possibility of also taking an oral test.
In the written test, the student must solve practical and theoretical exercises. The written test is passed
students who obtain a mark of 18/30 or higher. The maximum mark that
in the written test is 26/30.
The oral test is optional and is reserved for students who have passed the written test.
The oral test may raise, lower or leave unchanged the mark for the written paper.
theory questions (definitions, statements, demonstrations) and exercises. Students who intend to take
oral examination must necessarily attend the oral examination immediately following the written examination taken.
Once the written tests have been corrected, the professor will inform all those enrolled in the examination of the results of the test.

If the result of the test is positive (score greater than or equal to 18/30), the student has
2/3 days to decide whether to reject the written exam grade, accept the written exam grade or
request to take the oral test. The principle of silence-consent applies: if no communication has been received, the written test mark is recorded.
Failure to pass the
oral test or rejection of the mark awarded at the end of the oral test means that you must take the entire examination again.

Quality education.