MATHEMATICAL ANALYSIS II - channel 2

The goal of the course is to give students elements and fundamental techniques useful to study sequences and series of functions, calculate limits of functions of several variables, find maximum and minimum of functions, solve differential equations and systems, calculate integrals of functions of two or three variables, study the differential forms, calculate the integral of a differential forms.

1. (2 cfu). Sequences and series of functions. Pointwise and uniform convergence of sequences. Cauchy convergence criterion. Theorems of change of limits, continuity, derivability and integration. Series of functions. Pointwise, absolute, uniform and total convergence for series of functions. Power series. Theorem of Cauchy - Hadamard. Theorem of Abel. Series of Taylor. Sufficient conditions for series of Taylor. Fourier series. Sufficient conditions for the convergence of the Fourier series.

2. (2 cfu). Functions of n variable. Euclidean spaces. Functions in euclidean spaces. Limits of functions. Theorems regarding limits. Continuity. Continuous functions and connect subsets.Theorem of existence of the zeros. Continuous functions and compactness. Theorem of Heine-Borel.Theorem of Weierstrass. Uniform continuity and Theorem of Cantor. Lipschitz functions. Partial and directional derivatives. Differential functions. Differentiability and continuity. Theorem of the total differential. First differential. Derivatives and differential of upper order. Theorems of derivability of composite function. Theorem of Schwarz. Taylor's formula. Functions with zero gradient. Functions homogeneous. Theorem of Eulero. Local extremal points. Necessary and sufficient conditions for local extremal points. Implicit functions. Theorem of Dini. Systems of implicit functions.

    

3. (2 cfu). Differential equations. Differential equations of order n. Systems of n differential equations of first order. Equivalence between systems and equations. Cauchy problem. Solutions. Local and global existence and uniqueness for Cauchy problem. Linear systems. Set of solutions. Wronskiana matrix. Lagrange method. Systems with constant coefficients: basis of the space of solutions. Linear differential equations of order n. Euler equation. Non linear first order equations: equations with separable variables, homogeneous equations, Bernoulli equations.

4. (2 cfu). Measure and integration. Lebesgue measure. Measure of open, compact, bounded and unbounded sets. Properties of Lebesgue measure. Measurable functions.Lebesgue integral.Integration of bounded functions in subsets of finite measure. Integration of unbounded functions. Integration of functions in subsets of infinite measure. Sommability criteria.Theorem of B.Levi. Theorem of Lebesgue. Integration of series. Theorems of Fubini and Tonelli. Formulas of reduction for multiple integrals. Change of variables. Polar coordinates.

    

5. (1 cfu). Curves and differentail forms. Curves. Simple curves. Closed curves. Plane curves. Regular curves. Generally regular curves. Rectifiable curves and length. Curvilinear abscissa. Integral on a curve. Differential forms. Potential. Integral of a differential form. First integrability criterion. Closed differential forms and exact differential forms. Theorem of Poincar\'e. Simply connected open sets. Second integrability criterion. Gauss-Green formulas. Exact differential equations.

1. Di Fazio G., Zamboni P., Analisi Matematica 2, Monduzzi Editoriale.

2. Fanciullo M. S., Giacobbe A., Raciti F., Esercizi di Analisi Matematica 2, Medical Books.

3. D'Apice C., Durante T., Manzo R., Verso l'esame di Matematica 2, Maggioli editore.

4. D'Apice C., Manzo R., Verso l'esame di Matematica 3, Maggioli editore.