Identifying Data 2018/19
Subject (*) Mathematics I Code 631G02151
Study programme
Grao en Tecnoloxías Mariñas
Descriptors Cycle Period Year Type Credits
Graduate 1st four-month period
 First Basic training 6
Language
Spanish
Galician
English
Teaching method Face-to-face
Prerequisites
Department Matemáticas
Coordinador
Cao Rial, María Teresa
 E-mail
teresa.cao@udc.es
Lecturers
Campo Cabana, Marco Antonio
Cao Rial, María Teresa
Rodriguez Aros, Angel Daniel
 E-mail
marco.campo@udc.es
teresa.cao@udc.es
angel.aros@udc.es
Web http://www.nauticaymaquinas.es/
General description Nesta materia danse a coñecer os conceptos fundamentais e as aplicacións máis elementais de Álxebra Lineal, Xeometría do Plano e do Espazo Afín e Euclídeo, Análise de Funcións Reais dunha Variable Real e Variable Complexa. O alumno vai aprender a manexar con soltura as ferramentas básicas de Álxebra e Cálculo pero tamén a mellorar as súas habilidades na aprendizaxe e desenvolvemento de novos métodos e tecnoloxías necesarias para continuar a súa formación. Tamén a traballar con material bibliográfico e recursos informáticos, a elaborar unha memoria/informe de modo rigoroso e sistemático, a escribir e transmitir coñecementos correctamente, a realizar eficazmente as tarefas asignadas como parte dun grupo, etc. En concreto será capaz de resolver e analizar os resultados dos problemas matemáticos que poidan xurdir na enxeñería, a usar modelos matemáticos e a identificar o caso en que deben aplicarse.

Study programme competencies
Code Study programme competences
A12 CE12 - Interpretar e representar correctamente o espazo tridimensional, coñecendo os obxectivos e o emprego dos sistemas de representación gráfica.
A14 CE14 - Avaliación cualitativa e cuantitativa de datos e resultados, así como a representación e interpretación matemáticas de resultados obtidos experimentalmente.
A17 CE17 - Modelizar situacións e resolver problemas con técnicas ou ferramentas físico-matemáticas.
B1 CT1 - Capacidad para gestionar los propios conocimientos y utilizar de forma eficiente técnicas de trabajo intelectual
B2 CT2 - Resolver problemas de forma efectiva.
B3 CT3 - Comunicarse de xeito efectivo nun ámbito de traballo.
B4 CT4 - Traballar de forma autónoma con iniciativa.
B5 CT5 - Traballar de forma colaboradora.
B6 CT6 - Comportarse con ética e responsabilidade social como cidadán e como profesional.
B7 CT7 - Capacidade para interpretar, seleccionar e valorar conceptos adquiridos noutras disciplinas do ámbito marítimo, mediante fundamentos físico-matemáticos.
B8 CT8 - Versatilidade.
B9 CT9 - Capacidade para a aprendizaxe de novos métodos e teorías, que lle doten dunha gran versatilidade para adaptarse a novas situacións.
B10 CT10 - Comunicar por escrito e oralmente os coñecementos procedentes da linguaxe científica.
B11 CT11 - Capacidade para resolver problemas con iniciativa, toma de decisións, creatividade, razoamento crítico e de comunicar e transmitir coñecementos habilidades e destrezas.
C1 C1 - Expresarse correctamente, tanto de forma oral coma escrita, nas linguas oficiais da comunidade autónoma.
C3 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.
C5 C5 - Entender a importancia da cultura emprendedora e coñecer os medios ao alcance das persoas emprendedoras.
C6 C6 - Valorar criticamente o coñecemento, a tecnoloxía e a información dispoñible para resolver os problemas cos que deben enfrontarse.
C7 C7 - Asumir como profesional e cidadán a importancia da aprendizaxe ao longo da vida.
C8 C8 - Valorar a importancia que ten a investigación, a innovación e o desenvolvemento tecnolóxico no avance socioeconómico e cultural da sociedade.
C9 CB1 - Demostrar que posúen e comprenden coñecementos na área de estudo que parte da base da educación secundaria xeneral, e que inclúe coñecementos procedentes da vanguardia do seu campo de estudo
C10 CB2 - Aplicar os coñecementos no seu traballo ou vocación dunha forma profesional e poseer competencias demostrables por medio da elaboración e defensa de argumentos e resolución de problemas dentro da área dos seus estudos
C11 CB3 - Ter a capacidade de reunir e interpretar datos relevantes para emitir xuicios que inclúan unha reflexión sobre temas relevantes de índole social, científica ou ética
C12 CB4 - Poder transmitir información, ideas, problemas e solucións a un público tanto especializado como non especializado.
C13 CB5 - Ter desenvolvido aquelas habilidades de aprendizaxe necesarias para emprender estudos posteriores con un alto grao de autonomía.

Learning aims
Learning outcomes Study programme competences
A12
A14
A17
B1
B2
B3
B4
B5
B6
B7
B8
B9
B10
B11
C1
C3
C5
C6
C7
C8
C9
C10
C11
C12
C13

Contents
Topic Sub-topic
Lesson 1.- Vector Space 1.1.- Vector space. Definition. Examples and Properties
1.2.- Vector subspace.
1.3.- System of Generators of a Subspace
1.4.- Linear Independence
1.5.- Basis of a Vector Space. Finite Dimensional Spaces.
1.6.- Change of Basis in a Vector Space
1.7.- Union and Intersection of Subspaces
1.8.- Sum of Subspaces. Direct sum. Supplementary Subspaces.
1.9.- Product of Vectorial Spaces
Lesson 2.- Linear Functions. Matrices. 2.1.- Linear Function: Definition, Examples, Properties and Types of Linear Functions.
2.2.- Kernel and Image of a Linear Function.
2.3.- Existence and obtention of an Associated Matrix to a Linear Function.
2.4.- Addition of Linear Functions. Product by a Scalar. Associated Matrices.
2.5.- Vector Spaces of Matrices
2.6.- Composition of Linear Functions. Associated Matrix.
2.7.- Product of Matrices. Ring of Square Matrices
2.8.- Some Particular Types of Matrices
2.9.- Transpose Matrix. Symmetric, Antisymmetric and Orthogonal Matrices.
2.10.- Matrices of Complex Elements.
Lesson 3.- Determinants.
 3.0.- Permutations. Class of a Permutation.
3.1.- Determinant of a Square Matrix. Sarrus Rule.
3.2.- Properties of Determinants.
3.3.- Methods for Calculation of Determinants. Cofactor Matrix.
3.4.- Product of Determinants.
3.5.- Some Particular Examples of Determinants.
3.6.- Reverse Matrix.
3.7.- Rank of a Matrix.
3.8.- Rank of a System of Vectors
3.9.- Expression of the Change of Base of a Vectorial Space in shape Matrix
Lesson 4.- Systems of Linear Equations.
 4.1.- Definitions. Classification. Matrix notation.
4.2.- Equivalent systems.
4.3.- System of Cramer. Rule of Cramer
4.4.- General System of Linear Equations. Theorem of Rouché-Frobenius
4.5.- Homogeneous Systems.
4.6.- Methods of Resolution by Reduction. Gauss' Method.
Lesson 5.- Matrix Diagonalization.
 5.1.- Eigenvectors and Eigenvalues. Properties.
5.2.- Characteristic polynomial. Properties.
5.3.- Diagonalizable Matrices. Diagonalization.
5.4.- Diagonalization Of Symmetric Matrices.
Lesson 6.- Affine Space E3. Problems of Incidence and Parallelism.
 6.1.- Affine Space Associated to a Vector Space. System of Reference. Coordinates.
6.2.- Equations of Straight Lines.
6.3.- Relative positions of Straight Lines.
6.4.- Equations of a Plane.
6.5.- Relative positions of Planes. Bundles of Planes.
6.6.- Relative positions of Straight Lines and Planes.
Lesson 7.- Euclidean Vector Spaces. Scalar product, Vector product. Mixed Product. 7.1.- Scalar product
7.2.- Determination of a Scalar Product. Gram Matrix.
7.3.- Euclidean Vector Space. Definition.
7.4.- Norm of a Vector. Relevant Equalities and Inequalities.
7.5.- Angle of two Vectors. Orthogonality.
7.6.- Orthonormal Basis. Expression of the Scalar Product in an Orthonormal Basis.
7.7.- Euclidean Space E3.
7.8.- Orientation in E3.
7.9.- Vector product in R3 . Properties. Analytical expression.
7.10.- Mixed product. Analytical expression. Geometrical interpretation.
7.11.- Combined Products.
Lesson 8.- Metric Problems in Euclidean Spaces.
 8.1.- Normal equation of a Plane.
8.2.- Angles between Linear Manifolds in R3: Angle of Two Planes, Angle of Two Straight Lines, Angle of Straight Line and Plane.
8.3.- Distance between Linear Manifolds in R3: Distance of a Point to a Plane, Distance of a Point to a Straight Line. Distance between two Planes, Distance between Straight Line and Plane. Distance between two Straight Lines. Common Perpendicular to two Straight Lines.
8.4.- Cylindrical coordinates and Spherical coordinates in R3.
Lesson 9.-Real valued functions of a Real Variable. Continuity.
 9.1.- Basic definitions.
9.2.- Functional limits.
9.3.- Continuity. Types of Discontinuity.
9.4.- Properties and Theorems on Continuous Functions.
Lesson 10.- Differentiability and Applications of the Derivatives.

 10.1.- Derivative and Differential of a Function in a Point. Geometrical meaning.
10.2.- Properties and Calculation of Derivatives.
10.3.- Derivative function. Successive derivatives.
10.4.- Applications of the Derivatives to the Local Study of a Function: Growth and Decreasing. Maxima and Minima. Concavity and Convexity. Inflection points.
10.5.- Theorems of Rolle and Mean Value Theorem.
10.6.- Rules of L´Hôpital
Lesson 11.- Theorem of Taylor. Applications.

 11.1.- Expression of a Polynomial by means of his Derivatives in a Point.
11.2.- Polynomial and Theorem of Taylor. Formulae of Taylor and Mac Laurin.
11.3.- Expression of Lagrange for the Residual. Bounds for the residual.
11.4.- Applications to the Local Study of a Function: Monotonicity. Extremal values. Concavity and Convexity. Inflection points.
Lesson 12.- Graphic representation of Real Valued Functions. 12.1.- Domain and Continuity
12.2.- Symmetries
12.3.- Periodicity.
12.4.- Intersection with the coordinates axis.
12.5.- Use of successive derivatives and applications: Monotonicity. Extremal values. Concavity and Convexity. Inflection points.
12.6.- Asymptotes and Parabolic Branches
Lesson 13.- Sequences and Series.
 13.1.- General definitions. Types of Sequences.
13.2.- Practical calculation of Limits
13.3.- General definitions. Main Types of Numerical Series.
13.4.- Properties of the Numerical Series. Criteria of Convergence for Series of Positive Terms.
13.5.- Series of Positive and Negative Terms. Alternated Series.
Lesson 14.- Functional Sequences and Series. Series of powers.
 14.1.- General definitions.
14.2.- Series of Powers. Convergence.
14.3.- Series expansions.
14.4.- Series of Taylor and Mac Laurin.
14.5.- Binomial Series.
14.6.- Method of the Undetermined Coefficients.
Lesson 15.- Indefinite integration of Functions of a Real Variable
 15.1.- General definitions. Table of Primitives.
15.2.- Immediate integration
15.3.- Integration by Parts
15.4.- Integration of Rational Functions
15.5.- Integration by Replacement or Change of Variable
Lesson 16.- Definite Integration. Applications.
 16.1.- General definitions
16.2.- Properties
16.3.- Mean Value Theorem. Barrow's Rule.
16.4.- Evaluation of Definite Integrals.
16.5.- Improper Integral.
16.6.- Applications of the Definite Integral
Lesson 17.- Complex Numbers 17.1.- General definitions
17.2.- Fundamental operations
17.3.- Powers and Roots
17.4.- Exponential form of a Complex
17.5.- Logarithms And Complex Powers.
The development and overcoming of these contents, together with those corresponding to other subjects that include the acquisition of specific competencies of the degree, guarantees the knowledge, comprehension and sufficiency of the competencies contained in Table AIII / 2, of the STCW Convention, related to the level of management of First Engineer Officer of the Merchant Navy, on ships without power limitation of the main propulsion machinery and Chief Engineer officer of the Merchant Navy up to a maximum of 3000 kW. Table A-III / 2 of the STCW Convention.
Specification of the minimum standard of competence for Chief Engineer Officers and First Engineer Officers on ships powered by main propulsion machinery of 3000 kW or more.

Planning
Methodologies / tests Competencies Ordinary class hours Student’s personal work hours Total hours
Problem solving A12 A14 A17 B1 B2 B3 B4 B5 B7 B8 B9 B10 B11 C3 C6 C7 C8 C9 C10 C11 C12 C13 6 24 30
Guest lecture / keynote speech A12 A14 A17 B1 B2 B3 B4 B5 B6 B7 B9 C1 C3 C5 C7 C8 24 24 48
Objective test A12 A14 A17 B1 B2 B3 B4 B6 B7 B8 B10 B11 C1 C3 C5 C6 C8 4 0 4
Document analysis A12 A17 B1 B3 B4 B5 B7 B8 B9 B11 C3 0 2 2
Collaborative learning A12 A14 A17 B1 B2 B3 B5 B6 B7 B8 B9 B10 B11 C1 C3 C5 C6 C7 C8 9 9 18
Supervised projects A12 A14 A17 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 C1 C3 C5 C6 C7 C8 4 20 24
Online discussion A12 A14 A17 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 C1 C3 C5 C6 C7 C8 0 6 6
Directed discussion A12 A14 A17 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 C1 C5 C6 C7 C8 2 0 2
Diagramming A14 A17 B1 B2 B4 B5 B7 B8 B9 B10 C9 C11 C12 2 4 6
Introductory activities A12 A14 A17 B1 B4 B6 B7 B9 B10 C1 C3 C5 C6 C7 3 3 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
Problem solving En cada tema, vanse propoñer exercicios para resolver.
Guest lecture / keynote speech Exposición na aula dos conceptos fundamentais.
Objective test Proba de coñecementos.
Document analysis Seleccionar libros e páxinas web a utilizar
Collaborative learning Traballo en grupo con exposición dos resultados no seu caso
Supervised projects Traballos propostos individuais e grupais
Online discussion Plantexar e resolver dudas en Moodle
Directed discussion Discusión na aula do plantexado previamente en Moodle ou en clase.
Diagramming Facer esquemas
Introductory activities Tema 0: Conceptos básicos que se deben recordar

Personalized attention
Methodologies
Supervised projects
Collaborative learning
Guest lecture / keynote speech
Problem solving
Description
The students are encouraged to attend in small groups or individually to the professors' office to solve questions that may arise, thus obtaining a more specific guidance, acoording to their specific difficulties.

Assessment
Methodologies Competencies Description Qualification
Directed discussion A12 A14 A17 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 C1 C5 C6 C7 C8 Participación nos debates na aula.
Se avaliarán as competencias A12, A14, A17, B1, B2, B3, B5, B6, B7, B8, B9, B10, B11, C1, C3, C5, C6, C7 y C8. 		5
Supervised projects A12 A14 A17 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 C1 C3 C5 C6 C7 C8 Traballos propostos.
Se avaliarán as competencias A12, A14, A17, B1, B2, B4, B6, B7, B8, B9, B10, B11, C1, C5, C6, C7 y C8. 		10
Collaborative learning A12 A14 A17 B1 B2 B3 B5 B6 B7 B8 B9 B10 B11 C1 C3 C5 C6 C7 C8 Participación en traballos grupais.
Se avaliarán as competencias A12, A14, A17, B1, B2, B5, B6, B7, B8, B9, B10, B11, C1, C6, C7 y C8. 		5
Objective test A12 A14 A17 B1 B2 B3 B4 B6 B7 B8 B10 B11 C1 C3 C5 C6 C8 Comprobación dos coñecementos e capacidade de resolución de problemas.
Se avaliarán as competencias A12, A14, A17, B1, B2, B5, B6, B7, B8, B9, B10, B11, C1, C6, C7 y C8. 		70
Problem solving A12 A14 A17 B1 B2 B3 B4 B5 B7 B8 B9 B10 B11 C3 C6 C7 C8 C9 C10 C11 C12 C13 Resolver problemas.
Se avaliarán as competencias A12, A14, A17, B1, B2, B4, B5, B6, B8, B9, B10, B11, C1, C3, C6, C7 y C8. 		10
 
Assessment comments

The students that do not participate in the EHEA will be evaluated through a written test that will constitute 100% of the evaluation.  For those who participate in the EHEA, the written test will constitute 70% of the final marks.

Those students with recognition of part-time dedication and academic exemption of attendance, as established by the norm that regulates the regime of dedication to the study of undergraduate students in the UDC (Arts 2.3, 3.b, 4.3 e 7.5 ) (04/05/2017), and want to stay on the path of the EHEA and benefit from continuous assessment, must attend at least 50% of the course, exempting them from attending the theoretical classes, if they can not attend them. In case of not being able to attend the practical tests, they should attend tutorials at the proffesor office, where they will perform equivalent tests. 

The course is divided  in two parts: Part 1 (lessons 1-8) and part 2 (lessons 9-17). To pass it, it will be necessary to reach in each part a minimum of 3,5 points and afterwards obtain an average of, at least, 5 points.

In the unlikely case to reach  an arithmetic average of 5 but not having, at least, 3,5 points in each one of the parts, the result of the evaluation will be of fail and the final qualification will be calculated with a suitable geometric average.

The criteria of evaluation contemplated in the framewor A-III/1 and A-III/2 of the Code STCW and his amendments related with this matter have been taken into account for the design of this qualification methodology.

Sources of information
Basic Granero, F (). ALGEBRA Y GEOMETRÍA ANALÍTICA . Mac Graw-Hill
Fernández Viña, J.A (). ANÁLISIS MATEMÁTICO I . Tecnos
Granero, F. (). CÁLCULO . Mac Graw-Hill
García , A.y otros. (). CÁLCULO I (Teoría y Problemas) . Librería I.C.A.I
Granero, F. (). EJERCICIOS Y PROBLEMAS DE CÁLCULO (I y II) . Tébar Flores
D.G. Zill, W.S. Wright, J. Ibarra (). Matemáticas 1. Cálculo Diferencial. McGraw Hill
D.G. Zill, W.S. Wright, J. Ibarra (). Matemáticas 2. Cálculo Integral. McGraw Hill
S. Grossman, J. Ibarra (). Matemáticas 4. Álgebra Lineal. McGraw Hill
Á.M. Ramos del Olmo, J.M. Rey Cabezas (2017). Matemáticas básicas para el acceso a la universidad. Pirámide
Villa, A. de la (). PROBLEMAS DE ALGEBRA LINEAL. GLAGSA

Complementary


Recommendations
Subjects that it is recommended to have taken before

Subjects that are recommended to be taken simultaneously

Subjects that continue the syllabus

Other comments

Attend the optional introductory course the first week.


(*)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.