##
Yu.Burman

##
Calculus–3

- Abstract manifolds.
- Manifold, submanifold, smooth map.
- Smooth vector bundle, fiber bundle.
- The partition of unity. Tangent bundle via algebra of smooth functions.
- The weak theorem of Whitney (every n-dimensional manifold can be
embedded into
**R**^{2n+1}).

- Vector fields.
- Vector fields are derivations of the algebra of smooth functions.
- The commutator of vector fields.
- The Frobenius theorem about integrable distributions.
- Tensor operations over vector bundles. The behaviour of bundles and
their sections under maps.
- Riemannian metric and curvature.
Риманова метрика и кривизна.

- Differential forms.
- Differential forms and de Rham complex.
- Integration of the differential form over an oriented manifold.
- The Stokes theorem
- Riemannian metrics, the volume form and orientability.
- The Lie derivative of a vector field.
- The Frobenius theorem about integrable distributions.
- The Poincare lemma and other examples of computation of the de Rham
cohomology