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

The dates of the exams are below.

If you would like to see the marked sheets, we can send you the scanned PDF. Please request by email (in this case, attach a picture or a scan of your ID or the university booklet), or make an appointment and come to my office. Solutions will be availble after the oral exam.

1. 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.
2. The written test and the oral assessment must take place in the same session.
3. In sessions with two calls it is possible to sit for both written tests, but the delivery of the second one invalidates the first.
4. 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).
5. 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.
6. 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 29/1/18, 10:00, room B1 problems, solutions (corrected on 12 Feb 2018 (Problem 1)). 1st Oral assessment 5/2/18, 10:00, room B7
• Winter session 2nd Written test 21/2/18, 10:00, room B1 problems, solutions. 2nd Oral assessment 1-9/03/18 (due to suspeion caused by snow)
• Summer session 1st Written test 20/06/18, 10.00, room 2 problems, solutions (corrected on 13 September 2018 (Problem 1)). 1st Oral assessment 25/06/18, 10.00, room B6.
• Summer session 2nd Written test 09/07/18, 10.00, room A1, problems, solutions. 2nd Oral assessment 18/07/18, 10.00, room B5.
• Autumn session 1st Written test 30/08/18, 10.00, room A1, problems, solutions. 1st Oral assessment 03/09/18, 10.00, room 1122 of MATH DEPARTMENT.
• Autumn session 2nd Written test 17/09/18, 10.00, room 6, problems, solutions. 2nd Oral assessment 21/09/18, 10.00, room C4.

• 9/25 (Morsella) Sequences. Definition of limit. Operations with limits. Indeterminate forms.
• 9/27 (Morsella) Boundedness of convergent sequences. Monotonic sequences. Series. Definition of convergent, divergent, indeterminate series. Harmonic, telescopic and geometric series.
• 9/29 (Morsella) Exercises on sequences and series.
• 10/2 (Morsella) Necessary condition for convergence. Series with non-negative terms. Convergence tests: comparison, asymptotic comparison, integral, root and ratio. Generalized harmonic series.
• 10/4 (Morsella) Alternating series. Leibniz rule. Sum of the alternating harmonic series. Absolute and conditional convergence. Dirichlet's and Abel's tests. Examples.
• 10/6 (Morsella) Exercises on series.
• 10/9 (Morsella) Improper integrals. Definition. Comparison and asymptotic comparison tests (without proof). Absolute integrability (w.p.). Examples. Sequences of functions. Pointwise convergence. Examples.
• 10/11 (Morsella) Uniform convergence of sequences and series of functions. Examples. Continuity of the limit and passage to the limit under the integral. Weierstrass M-test.
• 10/13 (Morsella) Exercises on improper integrals and sequences of functions.
• 10/16 Power series, radius of convergence, term-by-term differentiation and integration.
• 10/18 Taylor's series for smooth functions, sufficient conditions for convergence of Taylor's series, solving differential equations by power series, binomial series.
• 10/20 Exercises on radius of convergence, limits and integration and differentiation, Taylor's series and differential equation.
• 10/23 Open sets, closed sets and boundaries in ℝⁿ, vector and scalar fields, their continuity.
• 10/25 Derivatives of scalar fields, directional, partial and total derivatives, gradients, differentiability of scalar fields.
• 10/27 Exercises on open sets, continuity, partial derivatives and gradients of scalar fields. Solutions (as there was a strike on public transport)
• 10/30 (Morsella) Chain rule for functions of several variables. Level sets. Tangent plane to the level sets and to the graph of a function. Differentiability of vector fields. Jacobian.
• 11/3 (Morsella) Examples and applications of the chain rule. Derivatives in polar coordinates. Derivatives of higher order. Schwarz's theorem (w.p.). Exercises on differentiability.
• 11/6 First order linear PDEs with constant coefficients, one-dimensional wave equation.
• 11/8 Implicit functions and their partial derivatives.
• 11/10 Exercises on PDEs and implicit functions. Solutions (as there was a general strike)
• 11/13 Minima, maxima and saddle points of scalar fields.
• 11/15 Lagrange's multipliers method, uniform boundedness of continuous scalar fields on a compact set.
• 11/17 Exercises on extrema and Lagrange's multipliers method.
• 11/20 Line integrals and their applications to mechanics (work, changes in kinetic energy, mass of a wire)
• 11/22 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.
• 11/24 Exercises on line integrals and gradients.
• 11/27 (Morsella) Methods for computing potentials. Differentiation under the integral sign. Potentials of vector fields on convex sets. Applications to differential equaations.
• 11/29 (Morsella) Partitions of rectangles. Step functions and their integrals. Notion of integrability for functions on a rectangle. Iterated integrals. Examples.
• 12/1 (Morsella) Exercises on vector fields and double integrals on rectangles.
• 12/4 (Morsella) Sets of content zero. Integrability of functions with discontinuity on sets of content zero. Notion of integrability for functions on bounded stes. Sets of Type I and II. Integrability of continuous functions on sets of Type I and II and reduction formulas.
• 12/6 Area and volume, mass, centroid and center of mass, Theorems of Pappus, Green's theorem and some applications, a sufficient and necessary condition for a vector field to be gradient.
• 12/11 Change of variables, polar coordinate, proof of the formula for change of variables, n-dimensional multiple integrals.
• 12/13 Parametrization of surfaces, vector products, area of a surface, examples (hemisphere in different parametrizations), another theorem of Pappus, surface integral, examples.
• 12/15 (Morsella) Exercises on double integrals.
• 12/18 Change of parametrizations of a surface, surface integral of vector fields, Stokes' theorem, curl and divergence and basic examles.
• 12/20 Further properties of curl and divergence, extensions of Green's theorem and Stokes' theorem to surfaces with holes, Gauss' theorem, another representation of divergence.
• Exercises on Green's theorem, surface integrals, Stokes' theorem and Gauss' theorem. Solutions (for self-study during the Christmas break)
• 01/08 Reviews and exercises on sequence, series and improper integrals.
• 01/10 Reviews and exercises on absolute/conditional convergence, power series and Taylor series.
• 01/12 Reviews and exercises on solving differential equations by power series, partial derivatives, chain rule, stationary points, partial differential equations.
• 01/15 Reviews and exercises on Lagrange's multiplier method, line integrals, conditions for a vector field to be a gradient, applications to exact differential equations.
• 01/17 Reviews and exercises on multiple integrals, Green's theorem, change of variables, polar/spherical/cylindrical coordinates.
• 01/19 Reviews and exercises on surface integrals, Stokes' theorem and Gauss' theorem.