| Study programme competencies |
| Code | Study programme competences |
| A1 | Capacidade para a resolución dos problemas matemáticos que poidan formularse na enxeñaría. Aptitude para aplicar os coñecementos sobre: álxebra lineal; xeometría; xeometría diferencial; cálculo diferencial e integral; ecuacións diferenciais e en derivadas parciais; métodos numéricos; algorítmica numérica; estatística e optimización. |
| A5 | Capacidade de visión espacial e coñecemento das técnicas de representación gráfica, tanto por métodos tradicionais de xeometría métrica e xeometría descritiva, coma mediante as aplicacións de deseño asistido por ordenador. |
| B1 | Que os estudantes demostren posuír e comprender coñecementos nunha área de estudo que parte da base da educación secundaria xeral e adoita encontrarse a un nivel que, aínda que se apoia en libros de texto avanzados, inclúe tamén algúns aspectos que implican coñecementos procedentes da vangarda do seu campo de estudo |
| B2 | Que os estudantes saiban aplicar os seus coñecementos ao seu traballo ou vocación dunha forma profesional e posúan as competencias que adoitan demostrarse por medio da elaboración e defensa de argumentos e a resolución de problemas dentro da súa área de estudo |
| B3 | Que os estudantes teñan a capacidade de reunir e interpretar datos relevantes (normalmente dentro da súa área de estudo) para emitiren xuízos que inclúan unha reflexión sobre temas relevantes de índole social, científica ou ética |
| B5 | Que os estudantes desenvolvan aquelas habilidades de aprendizaxe necesarias para emprenderen estudos posteriores cun alto grao de autonomía |
| B7 | Ser capaz de realizar unha análise crítica, avaliación e síntese de ideas novas e complexas |
| C1 | Utilizar as ferramentas básicas das tecnoloxías da información e as comunicacións (TIC) necesarias para o exercicio da súa profesión e para a aprendizaxe ao longo da súa vida. |
| C4 | Valorar criticamente o coñecemento, a tecnoloxía e a información dispoñible para resolver os problemas cos que deben enfrontarse. |
| C5 | Asumir como profesional e cidadán a importancia da aprendizaxe ao longo da vida. |
| Learning aims | |||
| Learning outcomes | Study programme competences | ||
| To think in a logic, critic and creative way. | B1 B2 B3 B5 B7 |
C4 C5 |
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| Ability of thinking in an abstract way, understanding and simplifying complex problems. | A1 |
B1 B2 B3 B5 B7 |
C1 C4 C5 |
| To understand the main characteristics of the formulation of a mathematical problem using the tools of the inifinitesimal calculus. | A1 A5 |
B2 B3 B5 B7 |
C4 |
| Get familiar with calculus language | A1 |
B1 B5 |
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| 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 be able to use the bibliography and the available IT tools to find the necessary information for solving a given problem | A1 A5 |
B5 B7 |
C1 C4 C5 |
| 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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| 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 B7 |
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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 B7 |
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| Contents |
| Topic | Sub-topic |
| The following topics develop the contents established in the verification report |
-Mean value theorems. -Introduction to vector calculus. -Taylor theorem and higher order derivatives. -Maximum and minimum. -Implicit function and inverse function. -Definite and indefinite integral. -Primitive Calculus. -Double and triple integrals. Applications to computing areas and volumes. |
| Complex numbers | The field of complex numbers. Operations: sum, produt. Module and argument. Polar form. Operating in polar form. |
| 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. |
| 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 B7 C4 C5 | 30 | 45 | 75 |
| Problem solving | A1 A5 B1 B2 B3 B5 B7 C4 C5 | 20 | 25 | 45 |
| Objective test | A1 A5 B1 B2 B3 B5 B7 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 | A1 A5 B1 B2 B3 B5 B7 C1 C4 C5 | Written exams to assess the knowledge of the subject by the students. The subject will consists on 3 parts and the final qualification will be the addition of the qualification obtained at each of these parts. 1) The first one will be done in the teaching period through a partial exam and will probably involve the chapters 1, 2, 3 and 4. 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). 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 90% of the final qualification. In case of passing any of these two parts, either in the partial or in the final exam of january, the qualification is retained for the present course until the exam of second oportunity. 3) The third part 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 second oportunity. |
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 | |