2019/2020 First semester (with O. Butterley)

Course description, time table: Mon 14:00-16:00, Wed 9:30-11:30, Fri 14:00-16:00, Aula 8

Prof. Morsella put some exercises with solutions on the topics he covered on his webpage.

Lecture notes (notes from 2018)Exercises (exercises from 2018)

- The exam consists of a written test (with Moodle) and a
**compulsory**oral test, both online. - To access the oral, a grade of at least
**18/30**must be obtained on the written test. The exam is passed if the final mark is at least 18/30. - The written test and the oral test must take place in the same
**call**. - Under penalty of exclusion, during written tests the use of electronic devices and applications except those required (MS Teams and possibly a web browser to access to
the Moodle quiz) is not allowed. It is
**not**possible to use books and notes. - It is necessary to have a webcam and a microphone to have test.

A simulation of the written test is available on the site.

- Winter session 1st Written test 24/1/20, 9:30, room C6, problems and solutions. 1st Oral assessment 29/1/20, 9:30, room C12.
- Winter session 2nd Written test 11/2/20, 9:30, room C6, problems and solutions. 2nd Oral assessment 18/2/20, 9:30, room C12.
- Summer session 1st Written test 30/06/20, 9:30 (online), problems and solutions. 1st Oral assessment 03/07/20, 9:30 (online).
- Summer session 2nd Written test 14/07/20, 9:30 (online), problems and solutions. 2nd Oral assessment 17/07/20, 9:30 (online).
- Autumn session 1st Written test 31/08/20, 9:30 (online), problems and solutions. 1st Oral assessment 31/08/20 (after written), 13:30 (online).
- Autumn session 2nd Written test 15/09/20, 9:30 (online), problems and solutions. 2nd Oral assessment 15/09/20 (after written), 13:30 (online).

- 9/23 Sequences of functions: pointwise and uniform convergence. Examples. Continuity of the limit and passage to the limit under the integral.
- 9/25 Series of functions: pointwise and uniform convergence. Continuity of the sum and term-by-term integration for uniformly convergent series. Weierstrass M-test. Power series. Examples. Existence of the radius of convergence. Differentiability of power series.
- 9/27 Exercises on sequences of functions and power series.
- 9/30 Taylor series. Unicity of the expansion in power series. Sufficient condition for the Taylor series expansion. Taylor series of sin x cosx, exp x. Solution of differential equations by power series. Binominal series.
- 10/2 Scalar and vector fields. Interior, exterior, boundary and accumulation points of sets in R
^{n}. Open and closed sets. Examples. Limit of a function of several variables. Operations with limits. - 10/4 Exercises on Taylor series, solutions of differential equations by power series, open sets and continuous scalar fields in R
^{n}. - 10/7 Directional and partial derivatives. Examples. Existence of partial derivatives does not imply continuity. Differentiability of scalar fields. Differentiability implies continuity and existence of directional derivatives. Sufficient condition for differentiability.
- 10/9 Curves and their parametrization, the tangent vector, Chain rule for scalar fields with examples. Level set of a function and examples.
- 10/11 Exercises on partial derivatives, gradients, chain rule and level sets.
- 10/14 Directional derivatives and differentiability of vector fields. Jacobian matrix. Chain rule. Examples and applications. Derivatives in polar coordinates. Derivatives of higher order. Commutativity of higher partial derivatives.
- 10/16 Solution of linear, first order partial differential equations with constant coefficients and of the one-dimensional wave equation.
- 10/18 Exercises on the chain rule and on partial differential equations.
- 10/21 Implicit functions and their partial derivatives.
- 10/23 Relative Minima, maxima and saddle points of scalar fields, Hessian matrix and a criterion for extrema.
- 10/25 Exercises implicit functions, stationary points and Hessian.

Exercises (exercises from 2018)

Home page of Yoh Tanimoto