Matemática III

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

Theoretical classes introduce concepts, properties and examples, enabling students to understand the exercises proposed and identify the mathematical tools to be used. In the remaining classes, shared, individual and/or group resolution of exercises and activities allows students to develop specific analysis and calculation skills to solve the problems proposed.

The course unit aims to develop students’ theoretical and applied understanding of fundamental mathematical concepts, essential for understanding the subjects covered in the other course units of the degree, particularly knowledge in the field of integral calculus in \mathbb{R}^n and numerical and power series. The key competencies to be developed include critical thinking, mathematical reasoning, and the ability to apply the acquired knowledge to problem-solving.

Theme 1 – Integral calculus in IR^n – Multiple integrals: double integrals (definition, properties and geometric interpretation, calculation in Cartesian and polar coordinates, applications); triple integrals (definition, properties, calculation of triple integrals, triple integrals in Cartesian, cylindrical and spherical coordinates, applications). Line integrals and vector calculus: vector functions (domain, limits, continuity and differentiability); curves and parameterisations; line integrals and their applications; the fundamental theorem of line integrals; Poincaré’s theorem; Green’s theorem.

Theme 2 – Series – Brief review of numerical sequences. Numerical series: definition; nature of a series; properties; examples (geometric series, Dirichlet series and telescopic series); necessary condition for convergence; series of non-negative terms; comparison criteria; integral, root and ratio criteria; absolute convergence and simple convergence; alternating series, Leibniz criterion. Power series: definition; radius and interval of convergence; properties; Taylor series, Taylor series expansions.

Continuous/periodic assessment - Assessment consists of two tests and assignments. The student is not allowed to use a mobile phone, calculator, or any other reference material unless expressly authorized by the course lecturer.

Components of continuous/periodic assessment:

• Test 1 - Worth 8 points (minimum of 2.0);
• Test 2 - Worth 8 points (minimum of 2.0);
• Assignments - Individual or group assignments, during classes, worth 4 points.

The final mark for continuous/periodic assessment corresponds to the sum of the marks for the assessment components, except when the student does not achieve the minimum mark in any of the assessments tests, in which case the final mark will be NRC (Not Approved).

In the normal period of exams, students may take the full exam (20 points, no minimum) or continue with continuous/periodic assessment and take one of the tests, applying the above-mentioned minimums.

In the appeal period, students must take the full exam (20 points, no minimum). During this assessment period, students who achieve a final mark of 9.0 may be asked to take an oral examination.

A student with a mark of 18 or above may be required to take an oral or written examination to defend their mark, should the lecturer so request. Supplementary oral examinations will be a form of assessment to be taken into account, whenever they are deemed necessary to clarify doubts and/or in cases where there is suspicion of cheating in the assessments carried out by the students.

NAO