Prerequisite: Analysis in Rn

Differentiable varieties, maple varieties, orientable varieties. Partition of unity. Application: Whitney’s dip theorem for compact varieties. Tangent and cotangent fibers. Differentiable applications, regular values. Alternating forms, differential forms, exterior differential. Surface integrals. Stokes theorem. De Rham cohomology. Mayer-Vietoris sequence. Homotopy invariance. Vector fields as sections and as derivations. Tensors. Applications.

References:
LIMA, E.L. – Analysis Course – Vol. 2. Rio de Janeiro, IMPA, Projeto Euclides, 1989.
LIMA, E.L. – Basic Homology. Rio de Janeiro, IMPA, Projeto Euclides, 2009.
SPIVAK, M. – Calculus on Manifolds. New York. Benjamin, 1965.
TU, L. W. – An introduction to manifolds. Universitext. Springer, New York, 2008.

* Basic syllabus. The teacher has the autonomy to make any changes.