Mathematics II

Differential and integral calculus of real functions of a real variable.

Presentation and analysis of the subject of the course unit in theoretical classes. Resolution and discussion of exercises with guidance from the teacher in theoretical-practical classes.

Acquisition of mathematics’ knowledge essential for understanding the subjects taught in the remaining curricular units of the degree, namely knowledge  of ordinary differential equations, numerical series and power series, real functions of several real variables and their derivatives and multiple integrals. Ability to apply this knowledge to solve problems. Development of critical thinking and reasoning skills.

1. Differential Equations –  Introduction and motivation. First order ordinary differential equations: first order linear equation, Bernoulli’s equation, separable variable equation, zero degree homogeneous equation.

2. Differential and Integral Calculus in IR^n – Conics and quadric surfaces. Brief notes on topology in R^n. Real functions of several real variables and their derivatives: domain, contour and graph of a two-variables function, limit and continuity, partial derivatives and higher order partial derivatives, differentiability, directional derivative and gradient vector, maxima, minima and saddle points, extrema with constraints, Lagrange’s multipliers method. Multiple integrals: double integral (definition, properties, geometric interpretation, calculation of the double integral in Cartesian and polar coordinates, examples of application), triple integral (definition, properties, calculation of the triple integral in Cartesian, cylindrical and spherical coordinates, examples of application).

3. Series – Brief revision about numerical sequences. Numerical series: definition, nature, properties, examples (geometric series, Dirichlet’s series and telescoping series), necessary condition for convergence, series of non-negative terms, comparison test, integral test, root and ratio test, conditional and absolute convergence, alternating series, Leibniz’s rule. Real power series: definition, radius and interval of convergence, properties, Taylor series expansions.

Students may choose two alternative assessment models: periodic assessment and final exam assessment. Each student will have to choose the model he wants immediately after taking the 1st test.

Periodic evaluation
Two tests will be carried out, each one quoted for 10 values. To pass the course, the student must
simultaneously have:

• Minimum classification of 3.25 values in each test;

• Sum of the ratings of the two tests greater than or equal to 9.5 values.

• Attended at least 60% of the total theoretical and theoretical-practical classes up to the date of each assessment test (this requirement may not apply to students with "student worker" status or students in other situations, which will be analyzed on a case-by-case basis. )

The 1st test covers the material of chapters I and a part of chapter II and will be held in person in April, in a date to be set later. The 2nd test corresponds to the remaining part of chapter II and the entire chapter III, and will be held on the date set for the regular season exam. In the exam of the appeal period, students included in the periodic assessment may choose to take an exam on the whole subject or a test corresponding to the 1st test or the 2nd test. The access to the appeal period requires a prior registration with the academic services. In the normal period, the student who opted for continuous assessment will only be able to take the 2nd test.

Assessment by final exam
In the exams of the normal, recourse and special periods, there will be exams on the entire subject, where a student who obtains a grade greater than or equal to 9.5 values succeeds in the course. In the special season exam there will be only one full exam. 

Students with a classification greater than or equal to seventeen points, or whose authorship of the tests is not clear, may be called for an oral test to defend their grade.

NAO