Identifying Data 2022/23
Subject (*) Mathematics I Code 631G01101
Study programme
Grao en Náutica e Transporte Marítimo
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
Arós Rodríguez, Angel Daniel
 E-mail
angel.aros@udc.es
Lecturers
Arós Rodríguez, Angel Daniel
Cao Rial, María Teresa
 E-mail
angel.aros@udc.es
teresa.cao@udc.es
Web http://www.nauticaymaquinas.es/
General description Nesta materia daranse 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 poden aplicarse.

Study programme competencies
Code Study programme competences
A2 Interpretar e representar correctamente o espazo tridimensional, coñecendo os obxectivos e o emprego de representación gráfica.
A8 Modelizar situacións e resolver problemas con técnicas ou ferramentas físico-matemáticas.
A9 Avaliación cualitativa e cuantitativa de datos e resultados, así como representación e interpretación matemática de resultados obtidos experimentalmente.
B1 Aprender a aprender.
B2 Resolver problemas de xeito efectivo.
B3 Aplicar un pensamento crítico, lóxico e creativo.
B4 Comunicarse de xeito efectivo nun ámbito de traballo.
B5 Traballar de forma autónoma con iniciativa.
B6 Traballar de forma colaboradora.
B7 Comportarse con ética e responsabilidade social como cidadán e como profesional.
B8 Aprender en ámbitos de teleformación.
B9 Capacidade para interpretar, seleccionar e valorar conceptos adquiridos noutras disciplinas do ámbito marítimo, mediante fundamentos físico-matemáticos.
B10 Versatilidade.
B11 Capacidade de adaptación a novas situacións.
B12 Uso das novas tecnoloxías TIC, e de Internet como medio de comunicación e como fonte de información.
B13 Comunicar por escrito e oralmente os coñecementos procedentes da linguaxe científica.
B14 Capacidade de análise e síntese.
B15 Capacidade para adquirir e aplicar coñecementos.
B16 Organizar, planificar e resolver problemas.
B17 Expresarse correctamente, tanto de forma oral coma escrita, nas linguas oficiais da comunidade autónoma
B19 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.
B22 Valorar criticamente o coñecemento, a tecnoloxía e a información dispoñible para resolver os problemas cos que deben enfrontarse.
B23 Asumir como profesional e cidadán a importancia da aprendizaxe ao longo da vida.
B24 Valorar a importancia que ten a investigación, a innovación e o desenvolvemento tecnolóxico no avance socioeconómico e cultural da sociedade.
C10 Que os estudantes saiban aplicar os coñecementos adquiridos e a súa capacidade de resolución de problemas en contornas novas ou pouco coñecidas dentro de contextos máis amplas (ou multidisciplinares) relacionados coa súa área de estudo

Learning aims
Learning outcomes Study programme competences
Write and share knowledge correctly. B4
B13
B17
Effectively perform assigned tasks as part of a group. B1
B2
B3
B6
B14
B15
B19
B22
B23
B24
 C10
To be able to solve and analyze the results of mathematical problems that may arise in engineering. A2
A8
A9
 B2
B3
B5
B9
B10
B11
B12
B16
 C10
To use mathematical models and identify the case in which they should be applied. A2
A8
A9
 B1
B2
B3
B7
B8
 C10
To know the fundamental concepts and applications of Linear Algebra, Affine and Euclidean Geometry, Mathematical Analysis of Real Functions of a Real Variable and Complex numbers. A2
A8
A9
 B1
B2
B3
B5
B8
B9
B11
B13
B14
B16
B22
To know the basic tools of Algebra and Calculus. A2
A8
A9
 B2
B3
B5
B9
B14
B15
B16
B17
To improve skills in learning and developing of new methods and technologies necessary to the following years of their career. B1
B2
B4
B7
B9
B10
B11
B14
B15
B19
B22
B23
B24
 C10
To work with bibliographic material and computer resources. B1
B3
B12
B19
B22
B23
B24
To prepare a report in a rigorous and systematic way. A9
 B13
B14
B15
B16
B17

Contents
Topic Sub-topic
Lesson 1.- Matrices and Determinants. 1.1.- Matrices. Operations with matrices.
1.2.- Determinants.
Lesson 2.- Vector spaces.
 2.1.- Introduction.
2.2.- Definition, examples and properties.
2.3.- Linear subspace.
2.4.- Linear dependence and linear independence.
2.5.- Generator systems.
2.6.- Bases. Dimension.
2.7.- Equations of a linear subspace.
2.8.- Range of a system of vectors.
Lesson 3.- Linear Mappings 3.1.- Introduction.
3.2.- Linear mappings.
3.3.- Matrix associated to a linear mapping.
3.4.- Change of basis matrix.
Lesson 4.- Systems of Linear Equations.
 4.1.- Introduction.
4.2.- Definition, examples.
4.3.- Existence and uniqueness of solution. Rouche-Frobenius theorem.
4.4.- Cramer's rule.
4.5.- Gauss and Gauss-Jordan 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.
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 and Applications. Plot of a real function.


 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. Plot of a real function.
Lesson 12.- Indefinite integration of Functions of a Real Variable 12.1.- General definitions. Table of Primitives.
12.2.- Immediate integration
12.3.- Integration by Parts
12.4.- Integration of Rational Functions
12.5.- Integration by Replacement or Change of Variable
Lesson 13.- Definite Integration. Applications. 13.1.- General definitions
13.2.- Properties
13.3.- Mean Value Theorem. Barrow's Rule.
13.4.- Evaluation of Definite Integrals.
13.5.- Improper Integral.
13.6.- Applications of the Definite Integral

Lesson 14.- Complex Numbers 14.1.- General definitions
14.2.- Fundamental operations
14.3.- Powers and Roots
14.4.- Exponential form of a Complex
14.5.- 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 AII / 2, of the STCW Convention, related to the level of management of chief mates of the Merchant Navy, on ships without gross tonnage limitation and Master up to a maximum of 500 GT. Table A-II / 2 of the STCW Convention.

Mandatory minimum requirements for certification of masters and chief mates on chief on ships of 500 gross tonnage or more.

Planning
Methodologies / tests Competencies Ordinary class hours Student’s personal work hours Total hours
Guest lecture / keynote speech A2 A8 B1 B2 B3 B4 B15 B22 C10 28 28 56
Problem solving A2 A8 A9 B2 B5 B6 B10 B11 B12 B13 B15 B16 B17 B19 C10 24 36 60
Supervised projects A2 A8 A9 B1 B2 B3 B4 B5 B6 B8 B9 B12 B13 B14 B15 B16 B17 B19 B22 B23 B24 C10 0 10 10
Seminar A2 A8 A9 B2 B5 B6 B10 B11 B12 B13 B15 B16 B17 B19 C10 0 10 10
Document analysis A2 A8 B1 B2 B3 B4 B5 B6 B8 B9 B10 B11 B12 B13 B14 B15 B16 B17 B19 B22 0 3 3
Introductory activities B1 B3 B4 B7 B12 B14 B15 B22 2 2 4
Objective test A2 A8 A9 B2 B4 B5 B11 B12 B13 B14 B16 B17 B19 B22 B23 C10 2 0 2
 
Personalized attention 5 0 5
 
(*)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 Exposition in the classroom of the fundamental concepts.
Problem solving In each topic, exercises will be proposed to solve.
Supervised projects Proposed individual and group projects.
Seminar Individual and / or very small group tutorships.
Document analysis Select books and web pages to use
Introductory activities Introdución á materia
Objective test Knowledge assessment.

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


Due to the health situation caused by COVID-19, meetings with students will preferably be held online (MS Teams).

Assessment
Methodologies Competencies Description Qualification
Problem solving A2 A8 A9 B2 B5 B6 B10 B11 B12 B13 B15 B16 B17 B19 C10 Resolver problemas.
 15
Guest lecture / keynote speech A2 A8 B1 B2 B3 B4 B15 B22 C10 Resolución de cuestións teóricas ou prácticas breves relacionadas cos contidos da sesión maxistral 10
Objective test A2 A8 A9 B2 B4 B5 B11 B12 B13 B14 B16 B17 B19 B22 B23 C10 Proba para amosar os coñecementos teóricos e prácticos adquiridos.
 60
Supervised projects A2 A8 A9 B1 B2 B3 B4 B5 B6 B8 B9 B12 B13 B14 B15 B16 B17 B19 B22 B23 B24 C10 Traballos propostos.
 15
 
Assessment comments

The students participants in the EHEA should attend a minimum of 80% of the lessons, being the continuous assessment of 40% of the final score. The other 60% of the score will be obtained from the partial tests that will take place throughout the term.

The students who have followed the continuous assessment but have not reached the 50% of the score through the partial tests will have a chance to reach it through a final test. This final test will include all topics of the term (the partial tests do not exclude topics)

The students who decide to not take part in the EHEA will be evaluated  with an objective test that includes an individual test of assimilation of practical-theoretical knowledge and problem solving. Pass a partial exams do not convalidate the corresponding lessons in case the student must do the final exam.  A student who does not do at least one of hte partial exams or a final exam will be qualified as Not Presented.

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 INDICATE SUCH CONDITION AT THE BEGINNING OF THE COURSE and attend at least 50% of the interactive lectures. In case of not being able to attend these sessions, they should attend tutorials at the proffesor office or by TEAMS, where they will perform equivalent tests. 

Fraudulent conduct in tests or activities, once verified, will cause a final mark of 0, invalidating any mark obtained in the in previous activities, as established in the current academic regulations at UDC.

Sources of information
Basic 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

Complementary


Recommendations
Subjects that it is recommended to have taken before

Subjects that are recommended to be taken simultaneously
Phisics/631G01103
Physics I/631G02153

Subjects that continue the syllabus
Mathematics II/631G01106

Other comments

Attend the optional introductory course which is given 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.