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Yu.Burman

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Calculus–2

The principal statements proved in the course:

1. Lebesgue measure.

- The null-set is nowhere dense.
- Lebesgue's criterion of Riemann integrability.
- The image of a null-set under a smooth map is a null-set. It may be not a
null-set if the map is only continuous.
- Change of variable in the Riemann integral.
- Countable union of Lebesgue measurable sets is measurable. The Lebesgue
measure is countably monotonic.
- Closed and open sets are Lebesgue measurable. An example of the set which
is not measurable.
- Measurable functions form an algebra closed under convergence almost
everywhere.
- Yegorov's theorem.
- The definition of the Lebesgue integral is sound.
- If a function is Riemann integrable then its Riemann integral is equal to
its Lebesgue integral.
- The integral equals the area under the graph. Fubini's theorem.
- The theorem abour bounded convergence.

2. Manifolds and the local analysis.

- The definition of a tangent space is sound.
- The change of variables is a diffeomorphism.
- The preimage of a regular value of a smooth map $f:M \to N$ is a smooth
subanifold $S \subset M$.
- The tangent bundle is a smooth manifold.
- The canonical form of a function near a noncritical point.
- The Hadamard's lemma.
- The Morse's lemma (the canonical form of a function near a nondegenerate
critical point).
- The Morse's parametric lemma.
- The quotient of a maximal ideal in the algebra of smooth functions by its
square is isomorphic to the tangent space.
- The Sard's lemma.

3. Fourier transform.

- Convolution is associative, commutative and commutes with translations
and differentitations.
- Pontryagin duals to
**R**, **Z** and U(1).
- Commutators of the Fourier transform with multiplication, translation and
differentiation.
- Smoothness of a function vs asymptotics of its Fourier transform.
- Direct and inverse Fourier transforms are mutually inverse.
- The Fourier transform maps convolution to the product and vice versa. The
Fourier transform is L
_{2} orthogonal.
- The Fourier series of a continuous periodic function converges.