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 some kinds of differential equations, calculate integrals of functions of two or three variables, study the differential forms, calculate the integral of a differential forms.
Blackboard lessons with related sessions of exercises.
(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 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. Partial and directional derivatives. Differential functions. Differentiability and continuity. Theorem of the total differential. First differential. Derivatives 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.
(2 cfu). Differential equations. Differential equations of order n. Cauchy problem. Solutions. Local and global existence and uniqueness for Cauchy problem. First order equations: equations with separable variables, homogeneous equations, linear equations, Bernoulli equations. Linear differential equations of order n. Wronskiana matrix. Lagrange method. Euler equation.
(2 cfu). Measure and integration. Riemann measure. Riemann integral. Integration of bounded functions. Formulas of reduction for multiple integrals. Change of variables. Polar coordinates. Guldino theorem.
(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.
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