| Study programme competencies |
| Code | Study programme competences |
| A1 | Skill for the resolution of the mathematical problems that can be formulated in the engineering. Aptitude for applying the knowledge on: linear algebra; geometry; differential geometry; differential and integral calculation; differential equations and in partial derivatives; numerical methods; algorithmic numerical; statistics and optimization |
| A5 | Have a capacity for the space vision and knowledge of the techniques of graphic representation, so much for traditional methods of metric geometry and descriptive geometry, as through the applications of design assisted by computer |
| B1 | That the students proved to have and to understand knowledge in an area of study what part of the base of the secondary education, and itself tends to find to a level that, although it leans in advanced text books, it includes also some aspects that knowledge implicates proceeding from the vanguard of its field of study |
| B2 | That the students know how to apply its knowledge to its work or vocation in a professional way and possess the competences that tend to prove itself by the elaboration and defense of arguments and the resolution of problems in its area of study |
| B3 | That the students have the ability to bring together and to interpret relevant data (normally in its area of study) to emit judgments that include a reflection on relevant subjects of social, scientific or ethical kind |
| B5 | That the students developed those skills of learning necessary to start subsequent studies with a high degree of autonomy |
| B6 | Be able to carrying out a critical analysis, evaluation and synthesis of new and complex ideas. |
| C1 | Using the basic tools of the technologies of the information and the communications (TIC) necessary for the exercise of its profession and for the learning throughout its life. |
| C4 | Recognizing critically the knowledge, the technology and the available information to solve the problems that they must face. |
| C5 | Assuming the importance of the learning as professional and as citizen throughout the life. |
| Learning aims | |||
| Learning outcomes | Study programme competences | ||
| To think in a logic, critic and creative way. | B1 B2 B3 B5 B6 |
C4 C5 |
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| Get familiar with calculus language | A1 |
B1 B5 |
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| To understand the main characteristics of the formulation of a mathematical problem using the tools of the inifinitesimal calculus. | A1 A5 |
B2 B3 B5 B6 |
C4 |
| To be able to use the bibliography and the available IT tools to find the necessary information for solving a given problem | A1 A5 |
B5 B6 |
C1 C4 C5 |
| To be able to evaluate the difficuylty of a problem and to choose the most suitable technique among the studied ones to carry on its solution. Have a good predisposition for problem solving | B3 |
C1 C4 C5 |
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| To know the underlying geometrical meaning of the studied mathematical formalism. To be able to represent sets in the plane and in the three dimensional space using different coordinates systems | A1 A5 |
B1 B2 |
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| To obtain a basic knowledge of functions of several variables: level sets, limits, continuity | A1 A5 |
B1 B2 B3 |
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| Ability of thinking in an abstract way, understanding and simplifying complex problems. | A1 |
B1 B2 B3 B5 B6 |
C4 |
| To understand the importance of partial derivatives and their relation to instantaneous variation of a magnitude (phisical, chemical, economical) and to asses their utility for the correct mathematical formulation of problems in engineering | A1 |
B2 B5 B6 |
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| To understand the meaning of integrals and their usage for the formulation of several problems in engineering. To know how to apply integral for the computation of areas of plane figures, areas of a surface of revolution and solid volumes. | A1 |
B2 B5 B6 |
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| Contents |
| Topic | Sub-topic |
| The space R^n | The vector space R^n. Scalar product: norms and distances. Classification of points and sets. Topology of R^n: bounded set, extrema. Coordinates systems: polar, cylindrical and spherical coordinates. |
| Functions of several variables | Scalar and vector functions. Level sets. Continuity. Continuity in compact sets. |
| Differenciation of funcions of several variables | Directional derivative. Partial derivatives: properties and practical computing. Differential map of a function. Gradient, relation with partial derivatives. Relation between the differential map and partial derivatives: jacobian matrix. Higher order partial derivatives. Introduction to vector calculus. |
| Applications of the differenciation of functions of several variables | Taylor polynomial for funcions of one and several variables. Critical points. Classification: Hessian matrix. Constrained optimization: dimensionality reduction, Lagrange multipliers method. Implicit function and inverse function theorems. |
| Integration of funcions of one variable | Riemann sums. Integrable functions. Integral Calculus Theorems: Mean Value Theorem, Fundamental Theorem and Barrow's rule. Primitive Calculus. Polinomial interpolation. Numerical integration. Compound Simpson's Rule. Application of integral calculus to computing arc lengths, volumes of revolution and surface areas of revolution. |
| Integration of functions of several variables | Double integrals. Triple integrals. Change of variable in double and triple integrals. Application of integral calculus to computing volume and mass of a solid body and its center of mass. |
| Complex numbers | The field of complex numbers. Operations: sum, produt. Module and argument. Polar form. Operating in polar form. |
| Appendix: The free software program, MAXIMA | Practical sessions with the free software program MAXIMA |
| Planning | ||||
| Methodologies / tests | Competencies | Ordinary class hours | Student’s personal work hours | Total hours |
| Guest lecture / keynote speech | A1 A5 B3 B5 B6 C5 C4 | 30 | 45 | 75 |
| Problem solving | A1 A5 B1 B2 B3 B5 B6 C4 C5 | 20 | 25 | 45 |
| Objective test | A5 A1 B1 B2 B3 B5 B6 C1 C4 C5 | 6 | 0 | 6 |
| Workshop | A1 B1 B2 B3 C1 C4 | 10 | 10 | 20 |
| Personalized attention | 4 | 0 | 4 | |
| (*)The information in the planning table is for guidance only and does not take into account the heterogeneity of the students. | ||||
| Methodologies |
| Methodologies | Description |
| Guest lecture / keynote speech | The course will be developed during the regular classes where the professor will explain the main concepts and results of the subject. |
| Problem solving | This classes are organiized in such a way that we practice how to solve the proposed problems. |
| Objective test | Three exams will be carried out during the course. The first one will be a partial exam where only some of the chapters will be considered. A final exam will be done at the end of the semester. Furthermore a computer exam will be carried out. |
| Workshop | Problems are solved assisted by the computer programm Maxima. |
| Personalized attention |
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| Assessment | |||
| Methodologies | Competencies | Description | Qualification |
| Objective test | A5 A1 B1 B2 B3 B5 B6 C1 C4 C5 | Written exams to assess the knowledge of the subject by the students. The subject will consist on four parts and the final qualification of the subject will be de addition of the qualification obtained at each of these parts 1) The first one will be performed in the reserved period for the partial exams (about the beginning of November), and will involve all the chapters studied until the celebration of the exam. If the student passes this exam, the qualification is retained until the end of the present course. This part will be recoverable in the final exam (second chance), to be held in July. 2) The second (and final) exam will be carried out in the period of final exams. It will involve the second part of the subject and a second chance to pass the first part. The weight of both exams will be the 80% of the final qualification. In case of passing any of these two parts, either in the partial of november or in the final exam of january, the qualification is retained for the present course until the exam of second oportunity of july. 3) The third part consists of evaluating the content of the issue "Complex Numbers", either by the work done throughout the semester, or with performing a specific exercise in the final objective test. The weight of this part is 10% of the final grade. 4) The third exam will consist on the evaluation the competences using the program MAXIMA, where the students must show their capacity for problem solving using the MAXIMA program. The weight of this third part will be the 10% of the final qualification. This part WILL NOT be recoverable, but the obtanined qualification will be kept until July. |
100 |
| Assessment comments | |||
| Sources of information |
| Basic |
García, A. et al. (2007). Cálculo I. Teoría y Problemas de Análisis Matemático en Una Variable. Madrid. Clagsa
García, A. et al. (2007). Cálculo II. Teoría y Problemas de Análisis Matemático en Varias Variables. Madrid. Clagsa
Burgos Román, Juan de (2007). Cálculo infinitesimal de una variable. Madrid. McGraw-Hill
Soler, M., Bronte, R., Marchante, L. (1992). Cálculo infinitesimal e integral. Madrid
García Castro, F., Gutiérrez Gómez, A. (1990-1992). Cálculo Infinitesimal. I-1,2. Pirámide. Madrid
Tébar Flores, E. (1977). Cálculo Infinitesimal. I-II. Madrid. Tébar Flores
Coquillat, F (1997). Cálculo Integral. Madrid. Tebar Flores
Spiegel, M. R. (1991). Cálculo Superior. Madrid. McGraw-Hill
Marsden, J., Tromba, A. (2010). Cálculo vectorial. ADDISON WESLEY
Larson, R., Hostetler, R., Edwards, B. (2013). Calculus. . Brooks Cole
Salas, L., Hille, E., Etgen, G. (2003). Calculus. vol I-II. Madrid. Reverté
De Diego, B. (1991). Ejercicios de Análisis: Cálculo diferencial e intergral (primer curso de escuelas técnicas superiores y facultades de ciencias). Madrid. Deimos
Varios (1990). Problemas de Cálculo Infinitesimal. Madrid. R.A.E.C. |
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| Complementary | |
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There are many interesting webpages that can help with this subject, here we cite just a few:
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| Recommendations | ||||
| Subjects that it is recommended to have taken before |
| Subjects that are recommended to be taken simultaneously |
| Subjects that continue the syllabus | ||||
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| Other comments | |