Academics

Analysis I

Sequences and Series of Functions: point-wise and uniform convergence; Weierstrass' test; continuity, integrability and differentiability of uniform limits; Arzela-Ascoli theorem; convolutions; Weierstrass' approximation theorem. Topology of the Euclidean Space Rⁿ: sequences in Rⁿ; Cauchy's criterion and Bolzano-Weierstrass' principle; continuous mappings; Borel-Lebesgue theorem; compact sets; connected sets, path connectedness, connected components. Real Functions of n-Variables: differentiable paths; partial and directional derivatives; successive derivatives; k-differentiable functions; Schwarz's theorem; the derivative as a linear approximation; chain rule; mean value theorem; Taylor's formula; critical points; implicit function theorem; Lagrange multipliers. Curvilinear integrals: differential forms of degree 1; integral of a form, of a vector field and of a function along a path; exact forms and closed forms; invariance of the integral of a closed form under homotopy.

References:
LIMA, E.L. - Análise no Espaço Rn. Ed. Univ. de Brasília. E. Blucher, 1970.
LIMA, E. L. - Curso de Análise. Vols. 1 e 2. Rio de Janeiro, IMPA, Projeto Euclides, 1989.
SPIVAK, M. - Calculus on Manifolds. New York. Benjamin, 1965.