### B1 Course. Smooth Manifolds and Observables

#### General Introduction

- Observation mechanism in classical physics.
- States of a classical system.
- Commutative algebras and their homomorphisms.
- Spectra of commutative algebras.
- Geometric, complete, and smooth algebras.
- Ideals of points and submanifolds.
- Localization of algebras to open subsets of the spectrum.
- Spectral theorem: equivalence of the algebraic and geometric definitions.
of smooth manifolds
- Smooth maps of manifolds and homomorphisms of smooth algebras.

#### First Order Differential Calculus on Manifolds Algebraically

- Tangent vectors and vector fields.
- Relative vector fields.
- The functor D.
- Behaviour of tangent vectors and vector fields with respect to smooth
mappings of manifolds.
- The flow of a vector field.
- Lie derivative of vector fields.
- The Lie algebra of vector fields.
- Covectors and differential forms.
- Tensors and their operations.
- The algebra of differential forms.
- Behaviour of differential forms and covariant tensors with respect to
differentiable mappings of manifolds.
- Lie derivatives of covariant tensors.
- Exterior differential and the de Rham cohomology.
- Cartan’s formula and homotopy property of de Rham cohomology.
- Distributions and the Frobenius theorem.
- Cohomological theory of integration.

#### References