The course aims to provide the basics concepts concerning the succession and the series of functions, ordinary differential equations, integration of differential forms along paths in open domain of the R^n space.
Expected learning outcomes
Capability to relate different aspects of the subject, and self-confidence in the use of the main techniques of Calculus.
Sequences and series of functions: pointwise and uniform convergence. Power series: the domain and the radius of convergence. Abel's theorem. Taylor's series. Functionals spaces connected with the uniform convergence. A fixed point theorem. Implicit functions: Dini's theorem in the scalar and vector cases. The inverse function theorem. Constrained optimization. First order differential equations: the Cauchy problem, existence and uniqueness of solutions. Differential equations of higher order. Linear equations. Curves in R^n: length, integration along curves. Differential forms, exactness and related results. Curves and differential forms. Conservative fields. Closed and exact forms. Potentials.
Prerequisites for admission
Analisi Matematica 1/2, Geometria 1/2 (these courses are not mandatory, but they are strongly recommended).
Lessons from the teacher and public solution of esercises from a second teacher. A tutor is made available to students.
-) G. Molteni, "Note del corso", available on the web page http://users.mat.unimi.it/users/molteni; -) G. Molteni, M. Vignati, "Analisi Matematica 3", Città Studi ed.; -) N. Fusco, P. Marcellini, C. Sbordone "Analisi Matematica due", Liguori ed.; -) C. Maderna, P.M. Soardi "Lezioni di Analisi Matematica II", Città Studi ed.; -) C.D. Pagani, S. Salsa "Analisi matematica, vol. 2", Masson ed.
Assessment methods and Criteria
Both a written and an oral exam. The student is elegible for the oral exam only when its written exam is positively evaluated. The written exam mainly deals with exercises, while the oral part of the exam is essentially a discussion about the theoretical results which are proved in lessons.