**Academic Year/course: 2017/18**

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**423 - Bachelor's Degree in Civil Engineering**

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**28705 - Mathematics applied to engineering II**

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Syllabus Information
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__Syllabus Information__

**Academic Year:**

**Subject:**

**Faculty / School:**

**Degree:**

**ECTS:**

**Year:**

**Semester:**

**Subject Type:**

**Module:**

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**5.1. Methodological overview**

**The learning process designed for this subject is based on the following:**

Strong interaction between the teacher/student. This interaction is brought into being through a division of work and responsibilities between the students and the teacher. Nevertheless, it must be taken into account that, to a certain degree, students can set their learning pace based on their own needs and availability, following the guidelines set by the teacher.

The current subject, *Matemática Aplicada a la Ingenería II*, is conceived as a stand-alone combination of contents, yet organized into three fundamental and complementary forms, which are: the theoretical concepts of each teaching unit, the solving of problems or resolution of questions, at the same time supported by other activities.

The organization of teaching will be carried out using the following steps:

**Theory Classes**:- Theoretical activities carried out mainly through exposition by the teacher, where the theoretical supports of the subject are displayed, highlighting the fundamental, structuring them in topics and or sections, interrelating them.
**Practical Classes**:- The teacher resolves practical problems or cases for demonstrative purposes. This type of teaching complements the theory shown in the lectures with practical aspects.
**Individual Tutorials**:- Those carried out giving individual, personalized attention with a teacher from the department. Said tutorials may be in person or online.

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**5.2. Learning tasks**

**The programme offered to the student to help them achieve their target results is made up of the following activities…**

Involves the active participation of the student, in a way that the results achieved in the learning process are developed, not taking away from those already set out, the activities are the following:

- Face-to-face generic activities:
- Theory Classes: The theoretical concepts of the subject are explained and illustrative examples are developed as support to the theory when necessary.
- Practical Classes: Problems and practical cases are carried out, complementary to the theoretical concepts studied.

- Generic non-class activities:
- Study and understanding of the theory taught in the lectures.
- Understanding and assimilation of the problems and practical cases solved in the practical classes.
- Preparation of seminars, solutions to proposed problems, etc.
- Preparation of the written tests for continuous assessment and final exams.

The subject has 6 ECTS credits, which represents 150 hours of student work in the subject during the trimester, in other words, 10 hours per week for 15 weeks of class.

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**5.3. Syllabus**

**Matemática Aplicada a la Ingeniería II programme:**

- Planar and spatial curves: Frenet frame; curvature and torsion.
- Functions of several variables. Limits and continuity.
- Partial derivatives and differential; the chain rule.
- Extrema. Constrained extrema: the method of Lagrange multipliers.
- Double integral; change of variables.
- Triple integrals.
- Line integral. Work and energy. Green's Theorem.
- Surfaces. Surface integrals; Stokes and Gauss Theorems.
- Ordinary Differential Equations: basic concepts, existence and uniqueness.
- Analytic solvability.
- Qualitative aspects: fixed points and linear stability.
- Numerical methods: Euler, Runge–Kutta.
- Higher orden ODE: Oscillators; resonance. Beam stability.
- Higher order numerical methods (FDM y FEM).
- Introduction to Partial Differential Equations: separation of variables; vibrations.

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**5.4. Course planning and calendar**

The dates of the final exams will be those that are officially published at http://www.eupla.es/secretaria/academica/examenes.html.

Week | Theme | Topic | Tests | Weight | Content |
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1 | 1 | Curves | |||

2 | 2 | Continuity | |||

3 | 3 | Differentiability | |||

4 | Extrema | 1st test | 5 | Dif./Cont. | |

5 | 4 | Multiple Integrals | 2nd test | 5 | Integrals |

6 | 5 | Line Integrals | |||

7 | 6 | Surface Integrals | 1st Exam | 40 | Several V. |

8 | 7 | ODE: Introduction, 1st order | |||

9 | Linear equations | 3rd test | 5 | 1st order ODE | |

10 | 8 | Linear stability | |||

11 | 9 | Numerical Methods | |||

12 | 10 | Oscillators, resonance | 4th test | 5 | ODE |

13 | 11 | Beam Stability | |||

14 | 12 | PDE: Introduction | |||

15 | Separation of variables | 2nd Exam | 40 | ODE, PDE |

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**5.5. Bibliography and recommended resources**

**Main resources**

- Subject presentations (available in the subject's Moodle webpage)
- Problem sheets (available in the subject's Moodle webpage)
- Symbolic calculus tool
`Maxima`

http://andrejv.github.io/wxmaxima/.

**Bibliography**

**BB**Bradley, Gerald L.. Cálculo de varias variables / Gerald L. Bradley, Karl J. Smith ; traducción, José Luis Vicente Córdoba ; revisión técnica, Pedro Paúl Escolano Madrid [etc.] : Pearson Educación, 2007**BB**Zill, Dennis G.. Ecuaciones diferenciales con problemas de valores en la frontera / Dennis G. Zill, Michael R. Cullen . - 5ª ed. México[etc.] : International Thomson Editores, cop. 200**BC**Apostol, Tom M.. Calculus. Vol.1, Cálculo con funciones de una variable, con una introducción al álgebra lineal / Tom M. Apostol. - 2ª ed. reimp. Barcelona [etc.] : Reverté, cop. 2002**BC**Apostol, Tom M.. Calculus. Vol.2, Cálculo con funciones de varias variables y álgebra lineal, con aplicaciones a las ecuaciones diferenciales y a las probabilidades / Tom M. Apostol. - 2ª ed., 7ª reimp. Barcelona, [etc.] : Reverté, D.L. 2002**BC**Burgos Román, Juan de. Cálculo infinitesimal de una variable / Juan de Burgos Román Madrid[etc.] : McGraw-Hill, D.L.1997**BC**Burgos Román, Juan de. Cálculo infinitesimal de varias variables / Juan de Burgos Román . - 2ª ed. Madrid [etc.] : McGraw-Hill/Interamericana, cop. 2008**BC**Larson, Ron. Cálculo 1 : De una variable / Ron Larson, Bruce H. Edwards ; revisión técnica, Marlene Aguilar Abalo … [et al.] ; [traducción: Joel Ibarra Escutia … (et al.)]. - 9ª ed. México [etc.] : McGraw Hill, cop. 2010**BC**Larson, Ron. Cálculo 2 : De varias variables / Ron Larson, Bruce H. Edwards ; revisión técnica, Marlene Aguilar Abalo … [et al.] ; [traducción: Joel Ibarra Escutia … (et al.)]. - 9ª ed. México [etc.] : McGraw Hill, cop. 2010**BC**Marsden, Jerrold E.. Cálculo vectorial / Jerrold E. Marsden, Anthony J. Tromba ; Versión en español Javier Páez Cárdenas ; Colaboración técnica Purificación González Sancho . - 4a. ed México [etc.] : Addison-Wesley Longman, 1998**BC**Simmons, George F.. Ecuaciones diferenciales : con aplicaciones y notas históricas / George F. Simmons ; con un capítulo sobre métodos numéricos de John S. Robertson ; traducción Lorenzo Abellanas Rapun . - 2a ed. Madrid [etc.] : McGraw-Hill, D.L. 1993

You can also view an updated reference list through the webpage of the UNIZAR Library: http://psfunizar7.unizar.es/br13/eBuscar.php?tipo=a