Università Roma Tre

The numbers of the paragraphs and of the theorems refer to the book of Chierchia or the book of Giusti (indicated with [G]).

WP means ‘without proof’.

1. Riemann integral in Rn Review of the Riemann integral in one dimension ([G], par.1.1.1). Rectangles
in R2, compact support functions, simple functions and their integral, function definition integrable according to Riemann in R2 (hence Rn).
Definition of measurable set ([G], Def. 12.3), a set is measurable if and only if its boundary has zero measurement ([G], Prop. 12.1 WP). Normal sets with respect to the Cartesian axes. A continuous function on a measurable set is integrable ([G], Teo. 12.1 WP). Fubini reduction theorem ([G], Teo 12.2 WP).
Formula of change of variable in integrals (WP). Polar, cylindrical, spherical coordinates. Examples: calculation of some barycenters and moments of inertia.

2. Regular curves.
Regular curves in R ^ n ([G], chap 15). Tangent versor.
Two equivalent curves traveled in the same direction have the same tangent versor ([G], Teo 15.1).
Length of a curve. It is greater than the displacement ([G], Teo 15.2).
Two equivalent curves have the same length ([G], Teo 15.3 WP)
Curvilinear integrals.





3. Surfaces, flows and divergence theorem.
Recalls on the vector product. Definition of regular surface ([G], Def. 15.4). Tangent plane and normal versor. Area of ​​a surface ([G], Def. 15.6). Examples: graphs of functions and rotation surfaces. Surface integrals. Flow of a vector field through a surface. Examples. Statement of the divergence theorem. Demonstration of the divergence theorem (for normal domains with respect to the three Cartesian axes, [G] formula 16.21).

4. Differential forms and work.
1-Differential forms (paragraph 10.4). Integral of a 1-differential form (work of a vector field), closed and exact forms. A form is exact if and only if the integral on any zero closed curve. Example of incorrect form closed.
Derived under the sign of integral (Prop. 5.47). Starry sets (Prop. 10.16); a closed form on a starred domain is exact.
Irrational and conservative fields, solenoidal and potential vector (on starry sets, Prop 10.19). The Green theorem in the plane (Teo 10.20) The Rotor theorem (Teo 10.21).

5. Series and sequence of functions

Series and sequence of functions (Ch.5.3): point, uniform and total convergence. Continuity of the limit, integration and derivation of uniformly convergent sequences of functions (WP). Power series (Ch. 5.4): convergence radius (Teo 2.1 and Teo 2.5). Taylor series examples of elementary functions.

6. Fourier series
Fourier series (Ch. 9.1), Fourier coefficients. Properties of Fourier coefficients, Bessel inequality, Riemann Lebesgue Lemma (Proposition 9.8 WP) Pointwise convergence of the Fourier series (Dini test, Lemma 9.12 WP). Uniform convergence in the case of C1 functions (Prop. 9.24). Equality of Parseval.

7 Matrix power series

Rudin, Principles of Mathematical Analysis, McGraw-Hill

The numbers of the paragraphs and of the theorems refer to the book of Chierchia or the book of Giusti (indicated with [G]).
The module consists of the proofs of the following results.

1. Riemann integral in Rn
A set is measurable if and only if its boundary has zero measurement ([G], Prop. 12.1). A continuous function on a measurable set is integrable ([G], Teo. 12.1). Fubini reduction theorem ([G], Teo 12.2).


2. Regular curves.

Two equivalent curves have the same length ([G], Teo 15.3)

3. Series and sequence of functions

Continuity of the limit, integration and derivation of uniformly convergent sequences of functions.

6. Fourier series
Properties of Fourier coefficients, Bessel inequality, Lemem of Riemann Lebesgue (Proposition 9.8) Pointwise convergence of the Fourier series (Dini test, Lemma 9.12).

Rudin, Principles of Mathematical Analysis, McGraw-Hill