1. General information
Course:
MATHEMATICAL INSTRUMENTS FOR ENGINEERING II
Code:
38305
Type:
BASIC
ECTS credits:
6
Degree:
345 - UNDERGRADUATE DEGREE PROGRAMME IN CIVIL ENGINEERING
Academic year:
2022-23
Center:
603 - E.T.S. CIVIL ENGINEERS OF CR
Group(s):
20
Year:
1
Duration:
C2
Main language:
Spanish
Second language:
Use of additional languages:
English Friendly:
Y
Web site:
Bilingual:
N
Lecturer:
CRISTINA SOLARES MARTINEZ
- Group(s):
20
Building/Office
Department
Phone number
Email
Office hours
Edificio Politécnico/2-D32
MATEMÁTICAS
3255
cristina.solares@uclm.es
It will be communicated at the beginning of the course.
2. Pre-Requisites
It is convenient that students have taken the subject “Mathematical Instruments for Engineering I” and “Mathematical and Computational Tools for Civil Engineering” .
3. Justification in the curriculum, relation to other subjects and to the profession
This subject is essential for the formation of an engineer. Different concepts related to functions with several variables will be studied, which will allow the student to solve engineering problems involving differentiation, integration, differential geometry and optimization.
4. Degree competences achieved in this course
| Course competences | |
|---|---|
| Code | Description |
| CE01 | Students can apply their knowledge in the practical solution of civil engineering problems, with capacity for the analysis and definition of the problem, the proposal of alternatives and their critical evaluation, choosing the optimal solution with technical arguments and with capacity of defense against third parties. |
| CE02 | Students have the ability to broaden their knowledge and solve problems in new or unfamiliar environments within broader (or multidisciplinary) contexts related to their area of study. Self-study ability, to undertake further studies with a high degree of autonomy |
| CE04 | Students have the ability to solve mathematical problems that may arise in engineering. Ability to apply knowledge of: linear algebra; geometry; differential geometry; differential and integral calculus; differential and partial derivative equations; numerical methods; numerical algorithms; statistics and optimization. |
| CE06 | Students have a basic knowledge of the use and programming of computers, operating systems, databases and software with engineering application. |
| CG01 | Students achieve general knowledge of Information and Communication Technologies (ICT). |
5. Objectives or Learning Outcomes
| Course learning outcomes | |
|---|---|
| Description | |
| Students know the fundamentals and applications of Optimization in the field of civil engineering. | |
| Students know how functions and data are approximated by means of power and Fourier series expansions and their applications. | |
| Students can handle functions of one and several variables including their derivation, integration and graphic representation. They know the fundamentals and applications of Differential and Integral Calculus. | |
| Students are able to express correctly both orally and in writing and, in particular, they can use the language of mathematics as a way of expressing accurately the quantities and operations in civil engineering. Students get used to teamwork and behave respectfully. | |
| Students use mathematical and computer tools to pose and solve civil engineering problems. | |
| The students are able to handle and know the concepts of differential geometry. | |
| Additional outcomes | |
| Description | |
| Apply the concepts of continuity, limit and derivation of functions of several variables to solve engineering problems. | |
| Understand multiple integrals and curvilinear integrals, as well as their applications in engineering. | |
6. Units / Contents
-
Unit 1: Functions of several variables
-
Unit 1.1: Real functions with several variables, definition. Limits of real functions with several variables. Geometric interpretation. Limits in one direction and consecutive limits. Continuity of real functions of several variables. Partial derivatives. Geometric interpretation. Partial derivatives of further order. Directional derivatives. Differential and gradient. Jacobian. Taylor series.
-
-
Unit 2: Extremes of Several Variables Functions
-
Unit 2.1: Maxima and minima of real functions of several real variables. Calculation of conditioned extrema: method of Lagrange multipliers. Applications.
-
-
Unit 3: Plane curves
-
Unit 3.1: Introduction and definition. Analytic expression. Tangent and normal to a curve. Length. Curvature. Evolutes.Locus. Applications.
-
-
Unit 4: Space curves
-
Unit 4.1: Introduction and definition. Analytic expression.Length. Versor and tangent straight line. Normal plane. Osculating plane. Curvature vector. Versor and principal normal line. Curvature, curvature centre and curvature radius. Versor and binormal straight line. Rectifying plane. Torsion. Torsion radius. Frenet frame. Applications.
-
-
Unit 5: Surfaces
-
Unit 5.1: Analytic expression. Tangent plane. Versor and normal straight line. Space curves that lie on a Surface. Cone and cylinder circumscribed. Curvature. Surface generation: Conical, cylindrical and revolution surfaces.
-
-
Unit 6: Line integrals. Potential function
-
Unit 6.1: Vector Analysis. Concept of line integral and properties. Calculation of a line integral. Concept of potential function. Calculation of the potential function. Existence condition. Independence of path. Applications: work calculation, mass calculation, area calculation, fluids study, etc.
-
-
Unit 7: Double integrals
-
Unit 7.1: Concept of double integral. Geometric interpretation. Properties of double integrals. Calculation of double integrals. Change of variables in double integrals. Green's theorem. Applications.
-
-
Unit 8: Area of a surface. Surface integral.
-
Unit 8.1: Area of a surface. Expression of the area in parametric coordinates. Surface integral. Stokes's theorem. Applications.
-
-
Unit 9: Triple integrals
-
Unit 9.1: Concept of triple integral. Properties of the triple integral. Calculation of triple integrals. Change of variables in triple integrals. The Divergence theorem. Applications.
-
7. Activities, Units/Modules and Methodology
| Training Activity | Methodology | Related Competences (only degrees before RD 822/2021) | ECTS | Hours | As | Com | Description | |
| Class Attendance (theory) [ON-SITE] | Lectures | CE01 CE04 CE06 CG01 | 1.46 | 36.5 | N | N | ||
| Class Attendance (practical) [ON-SITE] | Problem solving and exercises | CE01 CE04 CE06 CG01 | 0.54 | 13.5 | N | N | ||
| Problem solving and/or case studies [ON-SITE] | Problem solving and exercises | CE01 CE02 CE04 CE06 CG01 | 0.24 | 6 | Y | N | ||
| Final test [ON-SITE] | Assessment tests | CE01 CE02 CE04 | 0.12 | 3 | Y | Y | ||
| Study and Exam Preparation [OFF-SITE] | Combination of methods | CE01 CE02 CE04 CE06 CG01 | 3.6 | 90 | N | N | ||
| Individual tutoring sessions [ON-SITE] | Problem solving and exercises | CE01 CE02 CE04 CE06 CG01 | 0.04 | 1 | N | N | ||
| Total: | 6 | 150 | ||||||
| Total credits of in-class work: 2.4 | Total class time hours: 60 | |||||||
| Total credits of out of class work: 3.6 | Total hours of out of class work: 90 | |||||||
As: Assessable training activity Com: Training activity of compulsory overcoming (It will be essential to overcome both continuous and non-continuous assessment).
8. Evaluation criteria and Grading System
| Evaluation System | Continuous assessment | Non-continuous evaluation * | Description |
| Assessment of problem solving and/or case studies | 40.00% | 0.00% | It includes exercises and problems that the students will solve individually or in groups. Includes practical exercises in the computer room. |
| Final test | 60.00% | 100.00% | It includes the partial examinations, the ordinary and extraordinary examinations |
| Total: | 100.00% | 100.00% |
According to art. 4 of the UCLM Student Evaluation Regulations, it must be provided to students who cannot regularly attend face-to-face training activities the passing of the subject, having the right (art. 12.2) to be globally graded, in 2 annual calls per subject , an ordinary and an extraordinary one (evaluating 100% of the competences).
Evaluation criteria for the final exam:
-
Continuous assessment:The evaluation is made up of 2 partials, each one evaluated by means of 60%-exam grade and 40%-problem solving and/or case studies grade. The minimum mark required in the partial exams is 4 out of 10. The minimum mark to pass the ordinary call is 5 out of 10. The partial exams with a minimum mark of 4 are kept for the ordinary and extraordinary calls. The note in problem solving and/or case studies is saved for the ordinary and extraordinary calls.
All the assessable activities that have been carried out during the course are recovered. All the assessable activities that have been carried out during the course are no kept for the next year. -
Non-continuous evaluation:The student will have to do a global exam (100%). The global exam will include all the course content. To pass the course, the student must obtain at least a 5 out of 10.
Unless stated otherwise, continuous evaluation criteria will be applied to all students.
Anyone choosing non-continuous assessment must notify it to the lecturer within the class period of the subject. The option is only available if the student¿s participation in evaluation activities (from the continuous assessment) has not reached 50% of the total evaluation for the subject.
For the retake exam, the assessment type used for the final exam will remain valid.
Specifications for the resit/retake exam:
Same that in final exam.
Specifications for the second resit / retake exam:
The student will have to do a global exam that will include all the course and competences content. To pass the course, the student must obtain at least a 5 out of 10, which will constitute 100% of his/her grade.
9. Assignments, course calendar and important dates
| Not related to the syllabus/contents | |
|---|---|
| Hours | hours |
| Final test [PRESENCIAL][Assessment tests] | 3 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 25 |
| Individual tutoring sessions [PRESENCIAL][Problem solving and exercises] | 1 |
| Unit 1 (de 9): Functions of several variables | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 5 |
| Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] | 2 |
| Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises] | .75 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 9 |
| Unit 2 (de 9): Extremes of Several Variables Functions | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 5 |
| Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] | 2 |
| Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises] | .75 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 9 |
| Unit 3 (de 9): Plane curves | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 2.5 |
| Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] | 1 |
| Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 5 |
| Unit 4 (de 9): Space curves | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 4 |
| Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] | 1 |
| Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises] | .75 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 7 |
| Unit 5 (de 9): Surfaces | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 4 |
| Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] | 1 |
| Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises] | .75 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 7 |
| Unit 6 (de 9): Line integrals. Potential function | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 4 |
| Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] | 1 |
| Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises] | .75 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 7 |
| Unit 7 (de 9): Double integrals | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 5 |
| Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] | 2 |
| Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 9 |
| Unit 8 (de 9): Area of a surface. Surface integral. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 4 |
| Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] | 1.5 |
| Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 7 |
| Unit 9 (de 9): Triple integrals | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 3 |
| Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] | 2 |
| Problem solving and/or case studies [PRESENCIAL][Problem solving and exercises] | .75 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 5 |
| Global activity | |
|---|---|
| Activities | hours |
10. Bibliography and Sources
| Author(s) | Title | Book/Journal | Citv | Publishing house | ISBN | Year | Description | Link | Catálogo biblioteca |
|---|---|---|---|---|---|---|---|---|---|
| Anton, Howard | Calculus : a new horizon | John Wiley & Sons | 0-471-15306-0 | 1999 |
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| Aranda, Ernesto | Problemas de cálculo vectorial | Lulu.com | 978-1-4092-5048-7 | 2009 |
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| Bradley, Gerald L. | Calculo | Prentice-Hall | 84-8322-041-5 | 2001 |
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| Burgos Román, Juan de | Análisis matemático II (de varias variables) : 90 problemas | García-Maroto Editores | 978-84-935271-2-9 | 2007 |
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| Burgos Román, Juan de | Curvas y superficies : [Definiciones, Teoremas y Resultados] | García-Maroto | 978-84-936299-3-9 | 2008 |
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| Burgos Román, Juan de | Integración sobre curvas y superficies: teoremas de integrac | García-Maroto Editores | 978-84-936712-7-3 | 2009 |
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| Castellano Alcántara, J. | Cálculo matemático aplicado a la técnica | Proyecto Sur | 84-8254-995-2 | 2000 |
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| Castillo E., Conejo A.J., Pedregal P., García R., Alguacil N. | Formulación y Resolución de Modelos de Programación Matemática en Ingeniería y Ciencia | Universidad de Castilla-La Mancha | 84-600-9751-X | 2002 |
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| Estrada Castillo, Octavio | Cálculo vectorial y aplicaciones | Grupo Editorial Iberoamerica | 970-625-189-8 | 1999 |
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| Fong, Yuen | Calculus | Springer | 981-3083-01-8 | 1999 |
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| García A.,García F., Gutiérrez A., López A., Rodríguez G., Villa A. | Cálculo II | CLAGSA | 84-921847-0-1 | 1996 | |||||
| Gilbert Strang | Calculus | Wellesley-Cambridge Press | https://math.mit.edu/~gs/calculus/ | ||||||
| Granero Rodríguez, Francisco | Cálculo infinitesimal : una y varias variables | McGraw-Hill | 84-481-1740-9 | 1995 |
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| Gray, Alfred | Modern differential geometry of curves and surfaces with Mat | Chapman and Hall | 978-0-58488-448-4 | 2006 |
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| Herrero, Henar | Informática aplicada a las ciencias y a la ingeniería con Matlab | E. T. S. Ingenieros IndustrialesLibrería-Papelería | 84-699-3109-1 | 2009 |
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| Jeffery Cooper | A Matlab Companion for Multivariable Calculus | Academic Press | 0-12-187625-X | 2001 | |||||
| Jeffrey, Alan | Mathematics for engineers and scientists | Chapman & Hall | 0412621509 | 1996 |
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| Jon Rogawski | Cálculo de varias variables | Reverté | 9788429151749 | 2012 | |||||
| Kevin M. O'Connor | CALCULUS Labs for MATLAB | Jones and Bartlett Publishers, Inc. | 0-7637-3426-8 | 2005 |
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| Krasnov, Mijail Leontevich | Análisis vectorial: breve exposición del material teórico y | URSS | 5-354-01103-5 | 2005 |
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| Larson, Ron | Cálculo II de varias variables | McGraw-Hill | 970-10-5275-7 | 2006 |
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| Losada, Rodriguez, R. | Análisis Matemático | Ediciones Pirámide | 1978 | ||||||
| Marsden, Jerrold E. | Cálculo vectorial | Pearson Educación | 84-7829-069-9 | 2004 |
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| Mataix Plana, José Luis | Mil problemas de cálculo integral : [tercera parte] : deriv | Dossat 2000 | 978-84-89656-06-2 | 1996 |
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| O'NEILL, Barrett | Elementos de Geometria diferencial | Limusa | 968-18-0671-9 | 1982 |
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| Oprea, John | Differential Geometry and its applications | The Mathematical Association of America | 978-0-88385-748-9 | 2007 |
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| Pita Ruiz, Claudio de J. | Cálculo vectorial | Prentice-Hall Hispanoamericana | 968-880-592-7 | 1995 |
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| Spiegel, Murray R. | Cálculo superior | McGraw-Hill | 970-10-0065-X | 1993 |
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| Stein, Sherman K. | Cálculo y geometría analítica | McGraw-Hill Interamericana | 958-600-250-0 (o.c.) | 1995 |
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| Stewart, James (1941-) | Cálculo multivariable | Thomson Learning | 970-686-123-8 | 2003 |
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| Suárez Rodríguez, María del Carmen | Cálculo integral y aplicaciones con Matlab | Pearson | 84-205-4215-6 | 2004 |
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| Vera López, A. | Curso de geometría Diferencial: curvas y superficies | UNED | 1993 | ||||||
| Vladimir Rovenski | Modeling of curves and surfaces with Matlab | Springer | 2010 |