270 - guida - Facoltà di Ingegneria

GUIDA DELLO STUDENTE
Aims
(english version)
It is planned to give basic knowledge concerning functions of one or several real variables (continuity, differentiability and integration), ordinary
differential equations and sequences and series R^n-valued and their applications to concrete problems. Eventually, the student will have to
prove to have understood both theoretical and applicative parts of the course and to be able to use those methods to solve concrete problems.
Topics
The field of real numbers. Completeness, l.u.b. and g.l.b. of non empty subsets. Limits of sequences in R. Divergent sequences. Uniqueness
of limits. Compairison principle. Operations with limits. Few noticeable limits. Monotone sequences and their limits. The number e. Series in R.
Convergence, divergence and indeterminacy. Series with terms of constant sign. Convergence criteria. Absolute convergence. Leibnitz
theorem. Extension to sequences and series with values in R^n. Accumulation points. Functions from R to R and their limits. Right and left
limits. Algebra of limits. Compairison theorem, permanence of sign. Monotone functions and their limits. Landau symbols. The principle of
substitution of little-o. Algebra of little-o and big-O. Few noticeable limits. Continuous functions from R to R and their algebra. Zeri, Weierstrass
and intermediate values theorems and their consequences. Discontinuous functions. Continuity of the composition map. Continuity of
monotone functions and of the inverse function. Open, closed, connected and compact subsets of R^n. Continuous functions from R^n to
R^m. Continuity of components. Continuity of the composition map and consequences. Weierstrass, Zeri and intermediate values theorems
for functions of several variables. Derivative of a function from R to R. Differentiable functions. Differentiability rules. Differentiability and
continuity. The chain rule. Right and left derivative. The derivative of the inverse function. Max and min. Fermat, Rolle and Lagrange
Theorems and their consequences. De l'Hopital rule. Taylor and Mac Laurin approximation of functions. Estimate of the remainder in Peano,
Lagrange and Schlomilch form. Recognizing critical points with Taylor formula. Taylor formula and limits. Convex and concave functions.
Drawing the graph of a function. Differential calculus for functions from R^n to R^m. Directional derivative and differentiable functions. The
Jacobian matrix. The gradient theorem. The chain rule and rules of differentiability. The total derivative theorem. Continuity of differentiable
functions. Higher order derivatives. The Hessian matrix and Schwartz theorem. Max and min for a function from R^n to R. Necessary and
sufficient conditions. Riemann integral for real function of one real variable. Linearity and monotonicity of the integral. Mean value and
weighted mean value theorems. Fundamental theorem of calculus. Integration rules. Integration of rational functions and some irrational
functions. Integrable functions (continuous, monotone). Taylor formula and the remainder term in integral form. Indefinite integral. Improper
integrals. Comparaison criterium. Integrability of some elementary functions. Absolute integrability. Improper integrals and series. The integral
criterium. The field of complex numbers. Modulus and argument of a complex number. Inverse of a complex number. Power, root, exponential
and logarithm of a complex number.
Exam
Oral proof for those students passing a preliminary written proof
Textbooks
Official Text: Bertsch, Dal Passo, Giacomelli, Analisi Matematica, Mc Graw Hill. Other reference texts: 1 - Marcellini, Sbordone; Analisi
Matematica 1; Liguori 2 - Fusco, Marcellini, Sbordone; Analisi Matematica 2; Liguori 3 - Giusti, Analisi Matematica 1; Bollati Boringhieri 4 -
Giusti, Analisi Matematica 2; Bollati Boringhieri 5 - Bramanti, Pagani, Salsa, Analisi Matematica 1, Zanichelli 6 - Bramanti, Pagani, Salsa,
Analisi Matematica 2, Zanichelli
Tutorial session
Monday, 12:30-13:30
12