Yu.Burman

Calculus–3

1. Abstract manifolds.
   1. Manifold, submanifold, smooth map.
   2. Smooth vector bundle, fiber bundle.
   3. The partition of unity. Tangent bundle via algebra of smooth functions.
   4. The weak theorem of Whitney (every n-dimensional manifold can be embedded into R²ⁿ⁺¹).
2. Vector fields.
   1. Vector fields are derivations of the algebra of smooth functions.
   2. The commutator of vector fields.
   3. The Frobenius theorem about integrable distributions.
   4. Tensor operations over vector bundles. The behaviour of bundles and their sections under maps.
   5. Riemannian metric and curvature. Ðèìàíîâà ìåòðèêà è êðèâèçíà.
3. Differential forms.
   1. Differential forms and de Rham complex.
   2. Integration of the differential form over an oriented manifold.
   3. The Stokes theorem
   4. Riemannian metrics, the volume form and orientability.
   5. The Lie derivative of a vector field.
   6. The Frobenius theorem about integrable distributions.
   7. The Poincare lemma and other examples of computation of the de Rham cohomology