Mathematical Analysis 3

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.