2018/2019 First semester (with G. Morsella)

Course description, time table: Mon 14:00-16:00 + exercise session 16:00-18:00, 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 notesExercises

- The exam consists of a written test (with grades 0-30) and an optional oral assessment (with grades 0-5). The written test is passed if a grade of at least 15/30 is obtained. The exam is passed if an overall grade (written + oral) of at least 18/30 is obtained.
- The written test and the oral assessment must take place in the same session.
- In sessions with two calls it is possible to sit for both written tests, but the delivery of the second one invalidates the first.
- After an oral exam, either you withdraw the both written and oral exams, or the results will be confirmed (there is no option to withdraw the result of the oral exam only).
- Under penalty of exclusion, during written tests the use of cell phones and Internet-connectable electronic devices is not allowed. It is possible to use books, notes and electronic calculators.
- In order to take part in both the written and oral examinations it is necessary to bring an ID (including the university booklet).

- Winter session 1st Written test 29/1/19, 10:00, room C2, problems, solutions. 1st Oral assessment 7/2/19, 10:00, room C4.
- Winter session 2nd Written test 14/2/19, 10:00, room C2, problems, solutions. 2nd Oral assessment 28/2/19, 10:00, room C4.
- Summer session 1st Written test 24/06/18, 14.00, room 4 problems, solutions. 1st Oral assessment 01/07/19, 10:00, room C1.
- Summer session 2nd Written test 10/07/18, 14.00, room 2, problems, solutions (corrected, 07/10/19). 2nd Oral assessment 17/07/19, 10:00, room C2.
- Autumn session 1st Written test 29/08/19, 10.00, room 4, problems, solutions. 1st Oral assessment 05/09/19, 10:00, room C3.
- Autumn session 2nd Written test 12/09/18, 14:00, room 4, problems, solutions. 2nd Oral assessment 19/09/19, 10:00, room C3.

- 9/24 (Morsella) Sequences of functions: pointwise and uniform convergence. Examples. Continuity of the limit and passage to the limit under the integral.
- 9/26 (Morsella) 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.
- 9/28 (Morsella) Exercises on sequences of functions and power series.
- 10/1 (Morsella) Differentiability of power series. Taylor series. Unicity of the expansion in power series. Sufficient condition for the Taylor series expansion. Taylor series of sinx, cosx, ex, log(1+x), arctanx, (1+x)α.
- 10/3 (Morsella) Solution of differential equations by power series. Scalar and vector fields. Interior, exterior, boundary and accumulation points of sets in Rn. Open and closed sets. Examples. Limit of a function of several variables. Operations with limits.
- 10/5 (Morsella) Exercises on Taylor series, solutions of differential equations by power series and sets in R
^{n}. - 10/8 (Morsella) Composition of continuous functions. Examples. Derivative of a scalar field w.r.t. a vector. Mean value theorem. Directional and partial derivatives. Examples. Existence of partial derivatives does not imply continuity.
- 10/10 (Morsella) Differentiability of scalar fields. Differentiability implies continuity and existence of directional derivatives. Sufficient condition for differentiability. Chain rule for scalar fields.
- 10/12 (Morsella) Exercises on limits, continuity, partial derivatives and differentiability of scalar fields.
- 10/15 (Morsella) Directional derivatives and differentiability of vector fields. Jacobian matrix. Chain rule. Examples and applications. Derivatives in polar coordinates. Derivatives of higher order. Schwarz's theorem (without proof).
- 10/17 (Morsella) Solution of linear, first order partial differential equations with constant coefficients and of the one-dimensional wave equation.
- 10/19 (Morsella) Exercises on the chain rule and on partial differential equations.
- 10/22 Implicit functions and their partial derivatives.
- 10/24 Minima, maxima and saddle points of scalar fields.
- 10/26 Exercises implicit functions and stationary points.
- 10/29 Lecture was suspended due to weather alert.
- 10/31 Lagrange's multipliers method, uniform boundedness of continuous scalar fields on a compact set.
- 11/05 Line integrals and their applications to mechanics (work, changes in kinetic energy, mass of a wire).
- 11/07 The first and second fundamental theorems of calculus for vector fields, a characterization of a gradient field, a necessary condition for a vector field to be a gradient.
- 11/09 Exercises on Lagrange's multipliers method, line integrals and gradients.
- 11/12 Methods for computing potentials. Differentiation under the integral sign. Potentials of vector fields on convex sets. Applications to differential equations.
- 11/14 Partitions of rectangles. Step functions and their integrals. Notion of integrability for functions on a rectangle. Iterated integrals. Volume of a solid as a double integral.
- 11/16 Exercises on vector fields and double integrals on rectangles.
- 11/19 Integrability of functions with discontinuity on sets of content zero. Sets of Type I and II. Integrability of continuous functions on sets of Type I and II and reduction formulas.
- 11/21 Area and volume, mass, centroid and center of mass, Theorems of Pappus.
- 11/23 Exercises on double integrals, area and type I,II regions.
- 11/26 Green's theorem, simply connected regions, a sufficient and necessary condition for a vector field to be gradient.
- 11/28 Change of variables, polar coordinate, proof of the formula for change of variables.
- 11/30 Exercises on Green's theorem and change of variables.
- 12/3 (Morsella) Multiple integrals: step functions on n-rectangles and their integrals, integrable functions on n-rectangles,
sets of content zero, integrability of bounded functions with content zero discontinuities on n-rectangles,
bounded integrable functions on bounded sets.
Sets in R
^{3}projectable on coordinate planes and reduction formulas for triple integrals. Change of variables in multiple integrals. Cylindrical and spherical coordinates. - 12/5 (Morsella) Implicit, explicit and parametric surfaces. Fundamental vector product and its geometrical interpretation. Tangent plane to a parametric surface. Area of a parametric surface. Examples.
- 12/7 (Morsella) Exercises on triple integrals and parametric surfaces.
- 12/10 Parametrization of surfaces, vector products, area of a surface, examples (hemisphere in different parametrizations), another theorem of Pappus, examples.
- 12/12 Change of parametrizations of a surface, surface integral of vector fields, curl and divergence and basic examles.
- 12/14 Exercises on surfaces, fundamental vector products, area, surface integrals.
- 12/17 Stokes' theorem, an example and generalizations.
- 12/19 Gauss' theorem, an example, an interpretation of divergence and Maxwell's equations.
- 12/21 Exercises on Stokes' theorem and Gauss' theorem.
- 01/07 Reviews and exercises on sequence and series of functions, pointwise and uniform convergence, power series and applications to differential equations.
- 01/09 Reviews and exercises on Taylor expansion, partial derivatives, chain rule, partial differential equations (linear with constant coefficients and the one-dimensional wave equation).
- 01/11 Reviews and exercises on minima, maxima and suddle points of a scalar field, Hessian matrix, Lagrange's multiplier method and partial derivatives of implicit functions.
- 01/14 Reviews and exercises on line integrals, length of a curve, potential, and its applications to exact differential equations.
- 01/16 Reviews and exercises on multiple integrals, area and volume, change of coordinages and polar, cylindrical and spherical coordinates.
- 01/18 Reviews and exercises on parametrization of surfaces, surface integrals, Stokes' and Gauss' theorems.

Exercises

Home page of Yoh Tanimoto