Class Mathematical Analysis II

In this first-year curricular unit, the fundamentals of differential and integral calculus in IRn are introduced, along with initial notions of ordinary differential equations. This course constitutes an essential component of students' foundational mathematical training, contributing to the development of analytical reasoning, abstraction skills, and problem-solving abilities in multidimensional contexts. The knowledge acquired has cross-cutting applications in several areas of engineering and science, enabling students to understand and model complex phenomena and preparing them for progression to more advanced courses within the study programme, as well as for future applications in academic and professional contexts.

Not applicable

DIFFERENTIAL CALCULUS IN Rn . Study of multivariate scalar and vector functions. Graphic representations. Topological notions in Rn. Calculation of limits by definition, iterated limits and directional limits. Continuity. - Derivatives of scalar fields. Partial derivatives. Vector gradient. Directional derivatives. Differentials. Continuity. Derived from the composite function. Implicit derivation. - Derivatives of vector fields. - Extrema of functions of two and three variables, unconstrained and constrained. - Applications.   INTEGRAL CALCULUS IN Rn - Integrals with 2 and 3 variables.  - Vector field line integrals. Theorem of Green. - Surface integrals.  - Applications.   ORDINARY DIFFERENTIAL EQUATIONS (ODEs) Definitions. Integration of the main classes of first- and second-order ODEs.

In this course, active and innovative pedagogical methodologies will be adopted, centered on the student and promoting autonomous, participatory, and applied learning. A Flipped Classroom approach will be used, encouraging prior engagement with theoretical concepts so that class time can be devoted to the practical resolution of exercises, guided discussion, and clarification of doubts. Complementarily, Problem-Based Learning (PBL) strategies will be integrated through the analysis and resolution of problems, fostering critical thinking, mathematical reasoning, and the ability to apply the methods studied. Digital support tools such as Moodle, GeoGebra3D, and Octave will also be used. Whenever appropriate, collaborative activities and applied exercises will be proposed, reinforcing knowledge consolidation and students' autonomy in problem-solving.

- Anton, H., Bivens, I., & Davis, S. (2014). Calculus (10th ed., Vol. 2). Wiley. - Apostol, T. M (2004). Calculus (volume 2). Editorial Reverté. - Azenha, A. Jerónimo, M. A. (1995). Elementos de cálculo diferencial e integral em IR e IRn. Editora McGraw Hill. - Kreyszig, E., (1998). Advanced Engineering Mathematics (6th Edition). John Wiley & Sons. - Textos de apoio e coleções de exercícios fornecidos ao longo das aulas pela docente.