ISTITUZIONI DI MATEMATICHE

The course aims at the basic logical-mathematical training as the ability to understand hypothetical-deductive pathways and to provide useful calculation tools useful for solving any kind of problem.

Foundation. Recalls of elementary algebra. Elements of set theory. Set of Natural Numbers and Induction Principle. Set of Integers, Rational and Real numbers. Sets of limited real numbers. Set Extreme. Intervals.

Elements of linear algebra. Matrices, definitions, first properties and operations. Determinant of a matrix, its calculation techniques and properties. Minor and Complementary Minor. Rank of a matrix, its calculation techniques and properties. Reverse matrix. Linear systems. Replacement Method. Cramer's theorem. Rouché-Capelli theorem. Reduction or Gauss method. Vectors and vector algebra, dependence and linear independence between vectors, bases, scalar product, vector, mixed and double vector product.

Elements of analityc geoemtry. Reference System, Straight line equations. Parallelism and perpendicularity. Conic classification: circumference, ellipse, parabola and hyperbola.

One variable real function. Notion of function. Graph of a function. Monotone functions. Reverse functions. Composite functions. Elementary functions: absolute value, power function, exponential function, logarithmic function, trigonometric function. Injective, surjective, biunique functions. Inverse function. Limits. Comparison theorems. Notable limits. Limit uniqueness theorem *. Continuous functions. Points of discontinuity of a function. Composition of continuous functions. Theorem of the permanence of the sign. Theorem of the existence of the zeros. Theorem of the existence of intermediate values. Weierstrass theorem. Continuity criteria for inverse functions and monotonic functions. Continuity of elementary functions.

Differential calculus. Derivative of a function. Geometric meaning. Derivability and continuity. Rules for calculating derivatives. Operations with derivatives. Derivatives of compound functions and inverse functions. Derivatives of elementary functions. Derivatives of higher order. Angular points, cusps. Fermat, Rolle, Cauchy and Lagrange theorems. Maximum and minimum relative. Criteria of monotony. Concavity, convexity and inflections. Convexity criterion. De l'Hopital theorem *. Asymptotes. Study of the graph of a function.

Elements of Integral Calculus. Primitive of a function. Undefined integral. Integration methods for decomposition in sum, by parts, by substitution. Integrals of fractional rational functions. Integration by rationalization. Definite integral. Properties of defined integrals. Theorem of the media. Integral function. Fundamental theorem of integral calculus. Calculation of defined integrals. Calculation of areas. Improper integrals. Convergence criteria.

Ordinary Differential Equations. Generalities and definitions, Cauchy problem. Equations with separable variables, homogeneous equations. Linear equations of the first order. Differential equations of order n with constant coefficients. Bernoulli's equation.

Complex numbers. Operations, algebraic, trigonometric and exponential form, nesimal roots of complex numbers.

1. P. Marcellini C. Sbordone, Elementi di Analisi Matematica Uno, Liguori Editore.

2. P. Marcellini, C. Sbordone , Esercizi di Matematica Vol. 1 (Tomo 1, Tomo 2, Tomo 3, Tomo 4) e Vol 2. (Tomo 2), Liguori (2009).

3. M. Bramanti, C.D. Pagani, S. Salsa., Analisi Matematica 1, Zanichelli (2014)

4. A. Ratto, A. Cazzani, Matematica per le Scuole di Architettura, Liguori Editore, (2010)