Mathematical Analysis and Geometry (students A-L)
annual course
Prof. Maria Salvato








Set theory and hints of topology: properties of sets and related operations. Numerical sets: N, Z, Q, R. Cartesian reference of the line and the plane. Graphical representation of the line and parabola with axis parallel to the ordinate axis and related interrelations with the first and second degree equations and inequalities. Algebraic equations and inequalities of a higher degree per second. Logarithmic and exponential equations and inequalities.

Topology on the line: concept of interval, around a point, point of accumulation and isolated point. Limited sets: minimum and maximum, lower extreme and upper extremity of a set.

Real functions of a real variable. The elementary functions: whole and broken algebraic functions, circular functions and relative inverse ones. Logarithmic and exponential functions. Monotone functions. Reverse functions. Composite functions.

Limits of functions: Definition of limit of a real function of real variable: limit in a point of R, limit to infinity. Right limit and left limit. Limit operations. Indefinite forms. Theorem of the uniqueness of the limit, Theorem of the carabinieri. Infinitesimal and infinite.

Continuous functions. Definition and properties. Points of discontinuity. Theorem of the permanence of the sign. Theorem of existence of zeros and intermediate values. Weierstrass theorem.

With regard to the theorems on limits, theoretical proofs are not required but only justification on the graphical aspects.

Derivatives and differential. Definition of derivative in a point and of derivative in a range: geometric meaning (dim.). Theorem related to the continuity of derivable functions (dim.). Operations with derivatives. Derivatives of compound functions and inverse functions.

Application of derivatives. Analysis of the points in which the first derivative is canceled: conditions of existence of relative maximums, relative minimums, horizontal inflections.

Rolle's theorems (dim.), Lagrange's theorem (dim.). Monotony criteria for derivable functions (graphic justifications). Convexity criterion. Oblique inflections. Points of discontinuity of the derivative: angular points, cusps, vertical flexes. The theorems of L'Hospital.

Study of the graph of a function. Differential of a function of a variable and its geometric interpretation (dim.).

Integration according to Riemann. Rectangle and integral area defined. Definitions and properties of defined integrals. The theorem of the media (graphic justification).

Undefined integrals. Integral function. Fundamental theorem of integral calculus (dim.). Primitives and characterization of the primitives of a function (dim.). Fundamental report of the integral calculation (dim.). Primitives of basic elementary functions. A nod to the calculation of areas of flat figures.

Functions of two or more variables.

Overview of topology in R2. Functions of two or more variables. Overview of the concept of limit for functions of two variables (page 42). Weierstrass theorem, theorem of existence of intermediate values ​​(page 46).

Partial derivatives. Subsequent derivatives. Hessian Matrix. Schwarz's theorem). Differential of functions of two variables. Differential operators: gradient, divergence, rotor.

Differentiable functions. Tangent plane equation. The differential theorem. The derivation theorem of compound functions. Directional derivatives.

Geometric interpretation of the gradient vector. Relative maximums and minima: necessary condition of the first order, critical points, saddle points, Hessian matrix, Hessian determinant, sufficient condition of the second order.

Introduction to differential equations and Cauchy problem: ordinary differential equation of order n, equations in normal form, Cauchy problem for a first order equation in normal form. Linear differential equations: definition and general properties. Homogeneous linear differential equations with particular reference to the first and second order. Integrals of differential equations. Representation of the general integral of a linear differential equation.

Linear differential equations of the first order. General integral of the linear (complete) equations of the first order. Differential equations of the first order with separable variables. Linear differential equations of the second order homogeneous. Dependent, independent, and determinative Wronskian functions. General integral of homogeneous linear equations of the second order. Cauchy problem for a homogeneous linear differential equation of the second order. Characterization of the general integral of homogeneous linear equations of the second order with constant coefficients. Resolution of the complete linear equations of the second order with constant coefficients with the constant variation method.

Curvilinear integrals and differential forms

Flat curves, curves in space. Parametric equations. Simple curves. Closed curves. Regular curves. Length of a curve arc, geometric motivation. Curvilinear abscissa. Curvilinear integrals.

Linear differential forms. Curvilinear integral of a differential form and its properties. Exact differential forms: relationship between the differential of a function and the primitive of a differential form. Characterization of the primitives of a differential form in a connected open. Theorem of integration of the exact forms. Conservative vector fields: potential of a vector field.

Double integrals

Normal domains. Area of ​​a normal domain. Double integrals on normal domains. Reduction formulas for double integrals. Regular domains.

Notes on the Gauss-Green formulas in the plan. Divergence theorem. Stokes theorem.

Recommended tex

[1] A. Ventre – Matematica, Parte prima – Liguori Editori.

[2] F. Casolaro – Integrali – Ed. Zanichelli.

[3] N. Fusco – P. Marcellini - C. Sbordone: Elementi di analisi 2. Ed. Liguori.

[4] P. Marcellini – C. Sbordone: Esercitazioni di Matematica – I vol., parte I e II –  Ed. Liguori.

[5] A.Alvino- L.Carbone -G. Trombetti: Esercitazioni di Matematica- vol. I, parte I e II-Ed. Liguori.