1) Sequence and series of functions. Uniform and total convergence. Theorem of double limit. Uniform convergence and differentiability. A nowhere differentiable continuous function. Weierstrass approximation theorem. Space of continuous functions on compact sets. Equicontinouos and equibounded sets. Ascoli-Arzelà theorem
2) Complements on ordinary differential equations, Peano existence theorem, Extension of solutions. Qualitative study of differential equation. Applications to geometric and physical problems.
3) Sigma-algebra and measure. Measurable functions. Integral of positive functions. Monotone convergence theorem. Fatou’s lemma. Integrable functions. Dominate convergence theorem. Lebesgue measure in R and R^n. Measure on algebra and semi-algebra. Caratheodory exstension theorem. Distribution function of measures in R. Product measure. Fubini and Tonelli theorems. Integral depending on a parameter.
Outline of Gamma and Beta functions. Stirling formula.
4) Curves and surfaces. Length of a curve and surface area (formulas). Integration on curves. Differential forms. Integration of differential forms. Exact forms and closed forms. Necessary and sufficient conditions. Applications to differential equations. The Gauss-Green formula.