Identifying Data 2019/20
Subject (*) Calculus Code 730G03001
Study programme
Grao en Enxeñaría Mecánica
Descriptors Cycle Period Year Type Credits
Graduate 1st four-month period
First Basic training 6
Language
Spanish
Galician
Teaching method Face-to-face
Prerequisites
Department Matemáticas
Coordinador
Campo Cabana, Marco Antonio
E-mail
marco.campo@udc.es
Lecturers
Campo Cabana, Marco Antonio
Torres Miño, Araceli
E-mail
marco.campo@udc.es
araceli.torres@udc.es
Web http://campusvirtual.udc.es/moodle
General description Nesta materia estudiarase fundamentalmente cálculo diferencial e integral para funcións de varias variables. Para iso será necesario antes introducir certos conceptos topolóxicos e comprender as funcións de varias variables a través do seu dominio e conxuntos de nivel. O cálculo diferencial permitirá abordar conceptos como o plano tanxente e as series de Taylor, ademais de empregarse para o cálculo de extremos. O cálculo integral introducirase repasando a integración de funcións de unha variable para logo xeneralizar os conceptos relacionados a funcións e varias variables.
Contingency plan

Study programme competencies
Code Study programme competences
A1 FB1 - 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.
B1 CB01 - 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 CB02 - 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 CB03 - 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 CB05 - Que os estudantes desenvolvan aquelas habilidades de aprendizaxe necesarias para emprenderen estudos posteriores cun alto grao de autonomía
B7 B5 - Ser capaz de realizar unha análise crítica, avaliación e síntese de ideas novas e complexas
C1 C3 - 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 C6 - Valorar criticamente o coñecemento, a tecnoloxía e a información dispoñible para resolver os problemas cos que deben enfrontarse.
C5 C7 - Asumir como profesional e cidadán a importancia da aprendizaxe ao longo da vida.

Learning aims
Learning outcomes Study programme competences
Being able to solve mathematical problems with applications in engineering. Abilities in geometry and differential geometry A1
B1
B2
B3
B5
B7
C1
C4
C5
Abilities in differential and integral calculus. A1
B1
B2
B3
B5
B7
C1
C4
C5

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 and applications 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.
Taylor theorem for scalar functions.
Critical points. Classification.
Hessian matrix.
Conditioned extremes: dimension reduction, Lagrange multipliers method.
Implicit function theorem and inverse function theorem.


Integration of funcions of one and several variables 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.
Double integrals.
Triple integrals.
Variable change in double and triple integrals.
Application of integrals: calculation of areas and volumes.
Complex numbers The field of complex numbers.
Operations: sum, produt.
Module and argument.
Polar form.
Operating in polar form.

Planning
Methodologies / tests Competencies Ordinary class hours Student’s personal work hours Total hours
Guest lecture / keynote speech A1 B3 B5 B7 C4 C5 30 45 75
Problem solving A1 B1 B2 B3 B5 B7 C4 C5 26 39 65
Mixed objective/subjective test A1 B1 B2 B3 B5 B7 C1 C4 C5 6 0 6
 
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.
Mixed objective/subjective 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.

Personalized attention
Methodologies
Problem solving
Description
The contents of the course as well as the methodologies require that students work partly in an autonomous way. This may generate some questions that they can solve by using office hours as scheduled. In addition, homework will be guided by the lecturers of the course.

Students with recognition of part-time dedication and academic exemption from attendance may use office hours as a reference in order to follow the course and be advised on autonomous work.

Assessment
Methodologies Competencies Description Qualification
Mixed objective/subjective test A1 B1 B2 B3 B5 B7 C1 C4 C5 These consist on written exams to assess the knowledge of the course by the students. The exames will be divided into 2 parts and the final qualification will be the addition of the qualification obtained in each of them.

1) The first one will be done during the teaching period by means of a partial exam. It will likely involve contents of chapters 1, 2, 3 and 4. Students passing this exam, will not need to repeat the corresponding questions in the final exams. Otherwise, this part will be recoverable in the final exams.

2) The final exam will be carried out in the period of final exams. It will be include contents of the second part of the subject and a second chance to pass the first part.

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 the second oportunity.
90
Problem solving A1 B1 B2 B3 B5 B7 C4 C5 After the completion of a thematic block, small collections of representative exercises will be proposed for evaluation. 10
 
Assessment comments

Students with recognition of part-time dedication and academic exemption from attendance will be graded under the same conditions than other students, as explained above.


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.

Complementary

There are many interesting webpages that can help with this subject, here we cite just a few:

  • http://www.cds.caltech.edu/~marsden/books/Vector_Calculus.html

  • http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/

  • http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/index.htm

  • http://193.144.60.200/elearning/  


Recommendations
Subjects that it is recommended to have taken before

Subjects that are recommended to be taken simultaneously

Subjects that continue the syllabus
Linear Algebra/730G03006
Statistics/730G03008
Diferential Equations/730G03011
Reliability Statistics and Numerical Methods/730G03046

Other comments

In order to get a sustainable neighbourhood and attain the aim of action number 5: “Docencia e investigación saudábel e sustentábel ambiental e social” of the  "Plan de Acción Green Campus Ferrol", the homework of this course will attend to the following:
              •  Preferably, virtual homework will be used, when printing is not required.
              •  In the case that paper is needed, then:
                  -     No plastic materials will be used.
                 -      Printing will be done both sides.
                 -      Recycled paper will be used as possible.


         In general, a sustainable use of natural resources will be done. Moreover, ethic principles related to sustainability will be followed.


(*)The teaching guide is the document in which the URV publishes the information about all its courses. It is a public document and cannot be modified. Only in exceptional cases can it be revised by the competent agent or duly revised so that it is in line with current legislation.