Università Roma Tre

Sets of numbers: natural, integer, rational, real and complex numbers. Completeness axiom and continuum hypothesis. Representation of the real numbers on the line. Algebraic, trigonometric and exponential representation of complex numbers. Fundamental theorem of algebra.

Vectors: algebraic and geometric representation. Sum of vectors, scalar multiplication, linear combinations, bases. Scalar product and vector product. Cauchy-Schwarz inequality, triangular inequality. Lines on the Cartesian plane, lines and planes in space: conditions for parallelism and perpendicularity.

Linear algebra: matrices, sum of matrices, scalar multiplication, product of matrices, transposed matrix. Algebra of square matrices: trace, determinant, integer positive power, inverse matrix. Linear systems: matrix representation, method of the inverse matrix, Cramer theorem, homogeneous systems, rank of a matrix and Rouché-Capelli theorem. Eigenvalues and eigenvectors. Spectral theorem for real symmetric matrices. Linear transformations on the Euclidean plane: rotations.

Functions of a real variable. Injective, surjective and bijective functions. Composition of functions. Invertibility and monotonicity. Critical points: maxima, minima and inflection points. Symmetries: even and odd functions, and periodic functions. Graph of a function, operations on graphs.

Elementary functions and their properties. Linear functions, absolute value, power function, exponential, logarithm and trigonometric functions. Solution of inequalities. Applications.

Limit: definition and properties. Theorems on limits. Rules. Indeterminate forms, infinites and infinitesimals. List of limits. Bounded and divergent functions. Asymptotes. Continuous functions and points of discontinuity. Theorems on continuous functions and counterexamples: Weierstrass theorem, theorem ox existence of zeroes, theorem of the intermediate value.

Derivatives: incremental ratio and definition of derivative. Geometric interpretation and tangent lines to the graph. Derivatives of elementary functions. Differentiation rules. Points. Differentiation theorems: Fermat, Rolle, Lagrange, Cauchy. Criteria for monotonicity and convexity. De l'Hopital theorems. Approximation of functions with polynomials and Taylor's formula. Applications.

Graphs: qualitative study of the graph of a function.

Differential calculus for functions of several variables. Limits for morte variables. Continuity. Directional derivatives, partial derivatives, gradient and Hessian matrix. Maximum, minimum and saddle points. Vector fields: divergence and curl.

1. Dispense disponibili online.
2. Carlo Sbordone, Paolo Marcellini, Elementi di Calcolo, Liguori.
3. Carlo Sbordone, Paolo Marcellini, Esercitazioni di Matematica (prima parte e seconda parte), Liguori.

▪ Set theory (hints).
▪ Number sets: natural, integers, rational, real and complex numbers.
▪ Vectors and vector calculus.
▪ Matrices, linear algebra and linear systems.
▪ Real-valued functions of one real variable:
- elementary functions: powers, exponentials, logarithms, trigonometric functions;
- limits, derivatives, Taylor formula;
- qualitative study of the graph.
▪ Hints of differential calculus in higher dimensions:
- gradient, directional derivative, Hessian matrix;
- divergence, rotor.

Theory
• Notes of the lectures.
• P. Marcellini, C. Sbordone, Elementi di Calcolo (versione semplificata per i nuovi corsi di laurea), Liguori Editore (2016).
• D. Benedetto, M. Degli Esposti, C. Maffei, Matematica per le Scienze della Vita (III edizione), Casa Editrice Ambrosiana -Zanichelli (2015).

Exercises
• Notes of the lectures.
• P. Marcellini, C. Sbordone, Esercitazioni di Analisi di Matematica I (prima parte e seconda parte), Liguori Editore (2016).
• S. Salsa, A. Squellati, Esercizi di Matematica - Calcolo infinitesimale e algebra lineare, Zanichelli (2001).