Yu.Burman

Calculus–2

The principal statements proved in the course:

1. Lebesgue measure.

1. The null-set is nowhere dense.
2. Lebesgue's criterion of Riemann integrability.
3. 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.
4. Change of variable in the Riemann integral.
5. Countable union of Lebesgue measurable sets is measurable. The Lebesgue measure is countably monotonic.
6. Closed and open sets are Lebesgue measurable. An example of the set which is not measurable.
7. Measurable functions form an algebra closed under convergence almost everywhere.
8. Yegorov's theorem.
9. The definition of the Lebesgue integral is sound.
10. If a function is Riemann integrable then its Riemann integral is equal to its Lebesgue integral.
11. The integral equals the area under the graph. Fubini's theorem.
12. The theorem abour bounded convergence.

2. Manifolds and the local analysis.

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

3. Fourier transform.

1. Convolution is associative, commutative and commutes with translations and differentitations.
2. Pontryagin duals to R, Z and U(1).
3. Commutators of the Fourier transform with multiplication, translation and differentiation.
4. Smoothness of a function vs asymptotics of its Fourier transform.
5. Direct and inverse Fourier transforms are mutually inverse.
6. The Fourier transform maps convolution to the product and vice versa. The Fourier transform is L₂ orthogonal.
7. The Fourier series of a continuous periodic function converges.