I. To acquire technics and methods regarding inverse and implicit functions in R^n with applications to constrained problems.

II. To acquire a good knowledge of the concepts and methods in the classical integration theory on R^n, and, in particular, on curves and surfaces in R^3 with corresponding applications in Physics.

II. To acquire a good knowledge of the concepts and methods in the classical integration theory on R^n, and, in particular, on curves and surfaces in R^3 with corresponding applications in Physics.

Curriculum

Programme

1. Riemann integral in Rn Review of the Riemann integral in one dimension. Rectanglesin R2, compact support functions, simple functions and their integral, function definition integrable according to Riemann in R2 (hence Rn).

Definition of measurable set, a set is measurable if and only if its boundary has zero measurement. Normal sets with respect to the Cartesian axes. A continuous function on a measurable and integrable set. Fubini reduction theorem.

Formula of change of variable in integrals (without size). Polar, cylindrical, spherical coordinates. Examples: calculation of some barycenters and moments of inertia.

2. Regular curves.

Regular curves in R ^ n. Tangent versor.

Two equivalent curves traveled in the same direction have the same tangent versor.

Length of a curve. It is greater than the displacement.

Two equivalent curves have the same length.

Curvilinear integrals.

3. Surfaces, flows and divergence theorem.

Recalls on the vector product. Definition of regular surface. Tangent plane and normal versor. Area of a surface. 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.

4. Differential forms and work.

1-Differential forms. 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. Starry sets; a closed form on a starred domain is exact.

Irrational and conservative fields, solenoidal and potential vector (on starry sets). The Green theorem in the plane. The Rotor theorem.

5. Series and sequence of functions

Series and sequence of functions: point, uniform and total convergence. Continuity of the limit, integration and derivation of uniformly convergent sequences of functions. Power series: convergence radius. Taylor series examples of elementary functions.

6. Fourier series

Fourier series, Fourier coefficients. Properties of Fourier coefficients, Bessel inequality, Lemem of Riemann Lebesgue. Pointwise convergence of the Fourier series (Dini test). Uniform convergence in the case of C1 functions. Equality of Parseval.

Core Documentation

Analisi Matematica II, GiustiAnalisi Matematica II, Chierchia

Type of delivery of the course

4 hours of frontal teaching 2 of exercises and 2 of tutoring a week In the event of an extension of the health emergency from COVID-19, all the provisions that regulate the methods of carrying out the teaching activities will be implemented. In particular, the following methods will apply: live remote lesson and recording of the lesson itself.Type of evaluation

written test and subsequent oral testProgramme

1. Riemann integral in Rn Review of the Riemann integral in one dimension. Rectanglesin R2, compact support functions, simple functions and their integral, function definition integrable according to Riemann in R2 (hence Rn).

Definition of measurable set, a set is measurable if and only if its boundary has zero measurement. Normal sets with respect to the Cartesian axes. A continuous function on a measurable and integrable set. Fubini reduction theorem.

Formula of change of variable in integrals (without size). Polar, cylindrical, spherical coordinates. Examples: calculation of some barycenters and moments of inertia.

2. Regular curves.

Regular curves in R ^ n. Tangent versor.

Two equivalent curves traveled in the same direction have the same tangent versor.

Length of a curve. It is greater than the displacement.

Two equivalent curves have the same length.

Curvilinear integrals.

3. Surfaces, flows and divergence theorem.

Recalls on the vector product. Definition of regular surface. Tangent plane and normal versor. Area of a surface. 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.

4. Differential forms and work.

1-Differential forms. 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. Starry sets; a closed form on a starred domain is exact.

Irrational and conservative fields, solenoidal and potential vector (on starry sets). The Green theorem in the plane. The Rotor theorem.

5. Series and sequence of functions

Series and sequence of functions: point, uniform and total convergence. Continuity of the limit, integration and derivation of uniformly convergent sequences of functions. Power series: convergence radius. Taylor series examples of elementary functions.

6. Fourier series

Fourier series, Fourier coefficients. Properties of Fourier coefficients, Bessel inequality, Lemem of Riemann Lebesgue. Pointwise convergence of the Fourier series (Dini test). Uniform convergence in the case of C1 functions. Equality of Parseval.

Core Documentation

Analisi Matematica II, GiustiAnalisi Matematica II, Chierchia

Type of delivery of the course

4 hours of frontal teaching 2 of exercises and 2 of tutoring a week In the event of an extension of the health emergency from COVID-19, all the provisions that regulate the methods of carrying out the teaching activities will be implemented. In particular, the following methods will apply: live remote lesson and recording of the lesson itself.Type of evaluation

written test and subsequent oral test