Calculus is a subject under the Basic Training module of the first year of the Degrees in Electrical Engineering, in Industrial Electronics and Automation Engineering, and in Mechanical Engineering. It is an annual core subject.



This subject is one of the three from these degrees related to the study of mathematical concepts. Calculus supports the rest of the subjects of the degree since it includes a set of techniques that are used to carry out qualitative and quantitative analyses, make hypotheses, develop alternative strategies and analyze results.



On the other hand, applications specific to the grades in which this subject is lectured are studied, such as the study of circuits, which serve as the basis for the Fundamentals of Electrical Technology course, present in the three Degrees.



To follow this course, you must master basic mathematical operations, know how to solve linear equations and inequalities using the appropriate methods, and have a basic knowledge of differential and integral calculus.

SPECIFIC COMPETENCES MODULE: Basic Training



FB1. 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 algorithmic; statistics and optimization.



FB3. Basic knowledge of the use and programming of computers, operating systems, databases and computer programs with application in engineering.



LEARNING OUTCOMES OF THIS SUBJECT



RA1. To develop basic knowledge about differential and integral calculus, differential geometry, differential and partial derivative equations and numerical methods to apply it with judgement in understanding problematic situations typical of engineering.



RA2. To consistently use the procedural knowledge associated with scientific methodology in solving problematic situations related to infinitesimal calculus, both numerical, simulation or pencil and paper.



RA3. To carry out qualitative and quantitative analysis, formulate hypotheses, develop alternative strategies and analyze results.



RA4. To work with information corresponding to engineering processes related to infinitesimal calculus, analyze and correctly communicate ideas using oral, written, graphic and mathematical language.



RA5. To work efficiently in a group in order to make decisions in the development of the proposed tasks

INTRODUCTION TO INFINITESIMAL CALCULUS Concepts of real functions of real variables are reviewed. Representation of elementary functions in various coordinate systems along with the algebra of complex numbers are studied.



REAL FUNCTIONS OF REAL VARIABLES Continuity and differentiability problems are treated for real functions of real variables, both from a theoretical and a geometric point of view, while considering the relationships existing between these concepts. Besides, differential and integral calculus problems are addressed in one and several dimensions, paying special attention to practical applications of problematic engineering situations.



ORDINARY DIFFERENTIAL EQUATIONS The resolution of linear ordinary differential equations and systems of equations is studied. Their importance in engineering problems is highlighted. Initial condition problems are also introduced.



LAPLACE TRANSFORM The direct and inverse Laplace transform are defined with special emphasis on their properties to obtain the corresponding transforms of several functions, including interval-defined functions. Their adequate algebraic and frequency domain interpretations are essential to understand applications to engineering problems.

MASTER classes are used to present the topics established in the syllabus of the subject, as well as for the resolution of the doubts that the students raise in a systematic, orderly and complete way. This allows them to stimulate their interest through dialogue and increase their level of learning.



CLASSROOM PRACTICALS are the necessary complement for the assimilation of the concepts acquired in a master class, as they are focused on the practical development of problems, individually or in small groups. The use of active methodologies that achieve a more complete training, reinforcing and consolidating content is promoted.



In COMPUTER PRACTICALS, with the help of the Mathematica software, the methods and techniques learnt in practical classes to solve mathematical problems are applied. In particular, problems that arise in engineering and whose manual resolution is excessively "cumbersome" are proposed and solved.



The SEMINARS will be used to develop the transversal competence of teamwork. The possibility of distributing students in teams composed of a small but sufficient number of students will allow us to achieve the learning outcomes related to such competence.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 85
  • Mathematical software computer practicals (10%) + Seminars (5%): (%): 15

CONTINUOUS EVALUATION:



Students who take advantage of the continuous evaluation system must participate in ALL the assessable activities, which will take place throughout the course, and their grade will be the weighted sum of the grades obtained in the aforementioned activities. These activities are:



- Written tests (85%): The subject will be divided into four Thematic Modules:



--Module I (functions of one variable and two variables): In the first semester, 22%.

--Module II (indefinite, defined and double integrals): In the January exam period, 26%.

--Module III (complex numbers and first-order differential equations and higher-order linear equations): In the second semester, 17%.

--Module IV (higher-order differential equations and Laplace Transform): At the end of the second semester, 20%.



- Seminars: Evaluation of teamwork that will take place within the teaching period of the subject, 5%.



- Computer Practicals: The test will be carried out at the end of the second semester, 10%.



All those students who have participated in ALL the activities described above and for whom the weighted sum of ALL of the qualifications is greater than or equal to 5 out of 10 points will not have to attend the Ordinary Call.



Those who have participated in ALL the activities described above but have not achieved a weighted sum greater than or equal to 5 out of 10 points must sit the official exam in the Ordinary call if they want to obtain a grade. Otherwise, it is assumed that the student resigns from this call.



In the above-mentioned exam, the students will be able to resit the Thematic Modules I, II, III or IV of their choice to modify the previous mark, being able to keep the grades of the rest. Therefore, in the weighted sum corresponding to the definitive grade of the Ordinary Call, the last grades obtained in the repeated Thematic Modules, the grades obtained through the course for the non-repeated Thematic Modules, and the grades obtained through the course in Seminars and Computer Practicals, will be taken into account.





FINAL EVALUATION:



According to Article 8.3 of the regulations governing the evaluation of students in official degrees, students who submits in writing to the teaching staff responsible for the subject the waiver of continuous evaluation by week 18, counting from the beginning of the course, will have the right to be evaluated through the final evaluation system, regardless of whether they have participated in the continuous evaluation system or not.



Students who, for unjustified reasons, have not carried out any of the six assessable activities -Written Tests (4), Seminars or Computer Practicals- will also have to take this Final Evaluation.



The grade will consist of:



- Written test (90%)

- Computer Practicals Test (10%). This exam will be carried out within the teaching period of the subject, at the end of the second semester.



NON-PRESENTIAL ASSESSMENT:



If due to force majeure it is necessary to carry out the assessable activities remotely, the same percentages and conditions mentioned will be maintained.



RESIGNATION:



Except for STUDENTS WHO HAVE ALREADY PASSED THE SUBJECT THROUGH CONTINUOUS ASSESSMENT, failure to appear for the OFFICIAL EXAMINATION will mean automatic resignation from this call.



MATERIALS, MEANS AND RESOURCES ALLOWED



The student will be able to use the material provided by the teaching team for the development of each of the evaluation tests.

The evaluation of students who have not passed the subject in the ordinary call, regardless of the evaluation system they have chosen, will be carried out exclusively through the final evaluation system. This test will consist of:



- Written test (90%)



- Computer Practicals Test (10%)



Students who have already taken the computer practicals test in the ordinary call and obtained a grade equal to or greater than 5 out of 10 points will not have to resit that part.



RESIGNATION:



According to Article 12.3 of the regulations governing the evaluation of students in official degrees, failure to appear for the test set on the official exam date will mean automatic resignation from the call.



MATERIALS, MEANS AND RESOURCES ALLOWED



The student will be able to use the material provided by the teaching team for the development of each of the evaluation tests.

eGela virtual platform
Mathematica software

Basic bibliography

J. Burgos (1995): Cálculo infinitesimal de varias variables. Madrid: McGraw-Hill

Barragués, J., Cortajarena, I., y Zabala, M. (2010). Análisis matemático con soporte interactivo en Moodle. Madrid:

Prentice-Hall.

Barragués, J., Cortajarena, I., eta Zabala, M. (2010). Analisi matematikoa. Madrid: Prentice-Hall.

C.H. Edwards, D.E. Penney and D. Calvis (2022): Differential Equations and Boundary Value Problems: Computing and

Modeling, 6th edition. USA: Pearson Education

W. Kaplan (1996): Matemáticas avanzadas para estudiantes de ingeniería. México: Fondo Educativo Interamericano

R. Larson, R.P. Hostetler y B.H. Edwards (2006): Cálculo y Geometría Analítica. México: McGraw-Hill

R.K. Nagle y E.B. Saff (1998 Fundamentos de ecuaciones diferenciales. México: Addison Wesley Longman

N. Piskunov (1977): Cálculo diferencial e integral. Moscú: Mir

E. Martínez (1996): Ecuaciones diferenciales y cálculo integral. Aplicaciones y ejercicios. Bilbao: Servicio Editorial,

Universidad del País Vasco.

S.L. Salas, EG.J. Etgen, E. Hille (2007): Calculus: one and several variables. Hoboken: John Wiley & Sons.

In-depth bibliography

B.P. Demidovich (2001): 5000 problemas de análisis matemático. Madrid: Paraninfo
A. García, G. Rodríguez, S. Romero y A. de la Villa (2002): Cálculo II. Teoría y problemas de funciones de varias variables. Madrid: CLAGSA
E. Kreysig (1986): Matemáticas avanzadas para la ingeniería. México: Limusa
M.R. Spiegel (1970): Transformada de Laplace. Madrid: McGraw-Hill (Serie Schaum)
M.R. Spiegel (1976): Cálculo superior. McGraw-Hill, serie Schaum, 1976
G.B. Thomas, R.L. Finney, M.D. Weir y F.R. Giordano (2003): Cálculo con Geometría Analítica. Boston: Addison-Wesley
C.R Wylie (1982): Matemáticas superiors para ingeniería. M¿wxico: McGraw-Hill
D.G. Zill (2007): Ecuaciones diferenciales con aplicaciones de modelado. México: Thomson Learning

Web addresses

3Blue1Brown [https://www.youtube.com/3blue1brown]
Advanced Calculus for Engineers [https://ocw.mit.edu/courses/18-075-advanced-calculus-for-engineers-fall-2004/]
Ingeniaritzaren oinarri matematikoak [https://ocw.ehu.eus/course/view.php?id=517]
Cálculo infinitesimal [https://ocw.ehu.eus/file.php/105/calculo_infinitesimal/calculo_infi/Course_listing.html]
Calculus 1 [https://www.khanacademy.org/math/calculus-1]
Multivariable Calculus [https://www.khanacademy.org/math/multivariable-calculus]
Differential Equations [https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/]