Academic Year 2018/2019 - 2° Year

MAT/05 - 9 CFU - 1° Semester

**Knowledge and comprehension**. The course aims at conveying to the student the knowledge and comprehensions of the mathematical concepts in the program: sequence and series of functions, limits, derivatives and extrema of functions of several variables, differential equations and systems, Lebesgue theory of integration, curves and differential forms. The student must be able to express the above concepts by using the rigorous language of mathematics. Moreover, he must be able to illustrate them by means of simple examples.

**Application of knowledge.** The course aims at giving to the students the tools to solve simple to intermediate level exercises on: series of fucntions, extrema of functions of several variables, differential equations, double and triple integrals, differential forms. When faced with a simple problem the student must be able to recognize the corresponding theoretical framework and develop an autonomous reasonement in order to find the solution.

The course consists of blackboard lessons on the theoretical parts and subsequent problem sessions. Occasionaly, electronic devices might be used.

Remark: The proofs are not required for th theorems marked with a star (*)

1.Sequences and series of functions. (2 cfu). Real sequences of functions of one real variable. Pointwise and uniform convergence. Characterization of uniform convergence through the suprema sequence. Cauchy test of pointwise and uniform convergence. Limits exchange theorem*, continuity theorem, derivability theorem *, passage of limit under integral sign theorem. Series of real functions of one real variable. Pointwise and uniform convergence. Cauchy test. Absolute and total convergence. Weierstrass test. Comparison among various type of convergence. Theorems of: continuity, derivation and integration by series. Power series. Radius of convergence and related theorem. Cauchy-Hadamard theorem. Abel theorem*. Properties of the sum function of a power series. Taylor series. Conditions for the Taylor expansion. Important expansions (sinus, cosinus, exp, etc.). Fourier series. Sufficient conditions for the Fourier expansion*.

2. FUNCTIONS OF SEVERAL VARIABLES. (2 cfu). Euclidean spaces.Functions between euclidean spaces. Algebra of functions. Composition of functions and inverse function. Limitis of functions . in euclidean spaces. Theorems which characterize the limit by sequences and restrictions. Continuous functions. Continuous functions and connection. Zeros existence theorem. Compactness and continuous functions. Heine-Borel theorem *. Weierstrass theorem. Uniform continuity. Cantor theorem*. Lipschitz functions. Directional and partial derivatives of scalar functions . Differentiable functons. Necessary condtions for differentiability. First derivatives and differential. Derivability of a composition of functions. Higher order derivatives and differentials. Schwartz theorem.*. Second order Taylor formula. al primo e al secondo. Zero gradient theorem. Homogeneous functions and Euler theorem*. Local maximum and minimum for functions of several variables. Fermat theorem . Basic facts about quadratic forms and characterizations of their sign. Second order necessary condition. Second order sufficient conditions. Absolute extremum points search. Basic facts on convex functions. Implicit functions and implicit function theorem (by Dini) for scalar functions of two variables. Scalar and vector implict functions of several variables and related Dini theorems*.

3. DIFFERETIAL EQUATIONS. (2 cfu). First and n order differential equation Systems of n differential equations of first order in n unknown functions. Equivalence between systems and equations. Cauchy problem and definition of its solution. Local and global Cauchy theorem*. Sufficient condition for a function to be Lipschtz. Linear systems. Global solutions of linear systems and structure of the solution set. Wronskian matrix. Lagrange method. Constant coefficients linear systems: construction of a base in the solution space in the case of simple eigenvalues. Linear differential equations of higher order. Euler equation. Solution methods for some specific type of differential equation: separable variable equations, homogeneous equations, Linear equations of the first order. Bernoulli equations.

4. MEASURE AND INTEGRATION. (2 cfu). Basic facts about Lebesgue measure in R^n. Elementary measure of intervals and multi-intervals. Measure of bounded open and closed sets. Measurability for bounded and nonbounded sets. Properties: countable additivity numerabile additività*, monotonicity, upper and lower continuity*, subtractivity . Measurable functions. Basics on the Lebesgue integration theory in R^n: Integration of bounded functions on measurable set of bounded measure. Mean value theorem. Integration of arbitrary measurable functions defined on measurable sets . Geometric meaning of the integral. Integrability tests. Passage of limit under integral sign. Theorem of B.Levi*, Theorem of i Lebesgue*. Integration by series. Method of the invading sets*. Theorem of differentiation under integral sign*. Fubini theorem*. Tonelli theorem*. Reduction formulas for double and triple integrals. Change of variables in integrals*. Polar coordinates in the plane, Spherical and cylindrical coordinates in the space. Comparison between Riemann and Lebesgue integrals*:

5. CURVES AND DIFFERENTIAL FORMS. (1 cfu). Curve in R^n. Simple, plane and Jordan curves. Union of curves. Regular and generally regular curves. Change of parameter. Rectifiable curves. Rectifiabilitry of regular curves*. Curvilinear abscissa. Curvilinear integral. Concept of a differential form and its curvilinear integral. Exact differential forms. Integrability criterion. Circuit integral. Closed forms. Star shaped open sets. Poincaré Theorem *. Simple connected sets. Integrability criterion of simple connected sets *. Regular domains, Green formulas. *.Exact differential equations.

The foreign students who cannot read the italian textbooks can use the following textbook.

Calculus: A Complete Course, 9/E

Robert A. Adams, Christopher Essex, *University of Western Ontario*

ISBN-10: 0134154363 • ISBN-13: 9780134154367

©2018 • Prentice Hall Canada • Paper, 1168 pp

Published 19 Jun 2017 •

Chapters: 9, 12-13, 14, 18, 11