Identifying Data 2024/25
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
Cao Rial, María Teresa
 E-mail
teresa.cao@udc.es
Lecturers
Calvo Garrido, María Del Carmen
Cao Rial, María Teresa
 E-mail
carmen.calvo.garrido@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.

Competencies / Study results
Code Study programme competences / results
A54 RA1C-Write, explain and transmit the theoretical knowledge acquired both orally and in writing using scientific-technical language.
A55 RA2C-Identify and relate acquired knowledge to other disciplines
A57 RA4C-Collecting and interpreting relevant data
B30 RA7H-Applying critical, logical and creative thinking
B31 RA9H-Effectively solve practical problems associated with the subject by applying the knowledge acquired.
B32 RA10H-Know, analyse, synthesise and apply the contents, fundamental concepts and applications of the subject.
B33 RA11H-Develop both individual and group work
B34 RA12H-Handle bibliographic material and computer resources.
B35 RA13H-Handle with ease the tools, techniques, equipment and/or material/instrumental of each subject.
B36 RA14H-Use information and communication technology (ICT) tools necessary for the exercise of their profession and for lifelong learning.
C14 RA16X-Produce a report in a rigorous and systematic way.

Learning aims
Learning outcomes Study programme competences / results
RA1C-Write, explain and transmit the theoretical knowledge acquired both orally and in writing using scientific-technical language. A54
RA2C-Identify and relate acquired knowledge to other disciplines A55
RA4C-Collecting and interpreting relevant data A57
RA7H-Applying critical, logical and creative thinking B30
RA9H-Effectively solve practical problems associated with the subject by applying the knowledge acquired. B31
RA10H-Know, analyse, synthesise and apply the contents, fundamental concepts and applications of the subject. B32
RA11H-Develop both individual and group work B33
RA12H-Handle bibliographic material and computer resources. B34
RA13H-Handle with ease the tools, techniques, equipment and/or material/instrumental of each subject. B35
RA14H-Use information and communication technology (ICT) tools necessary for the exercise of their profession and for lifelong learning. B36
RA16X-Produce a report in a rigorous and systematic way. C14

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 / Results Teaching hours (in-person & virtual) Student’s personal work hours Total hours
Guest lecture / keynote speech A55 A57 B30 B32 28 28 56
Problem solving A54 B30 B31 B32 B33 B35 B36 24 36 60
Supervised projects A54 A57 B30 B31 B32 B34 B35 B36 C14 0 10 10
Seminar A54 A55 B30 B31 B32 B33 B34 B35 0 10 10
Document analysis A55 A57 B34 B35 B36 0 3 3
Introductory activities B1 B3 B4 B7 B12 B14 B15 B22 2 2 4
Objective test A54 B30 B31 B32 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.


Meetings with students will preferably be held online (MS Teams).

Assessment
Methodologies Competencies / Results Description Qualification
Problem solving A54 B30 B31 B32 B33 B35 B36 Solve problems. 15
Guest lecture / keynote speech A55 A57 B30 B32 Resolution of theoretical or brief practical questions related to the contents of the guest lecture. 10
Objective test A54 B30 B31 B32 Test to show the theoretical and practical knowledge acquired. 60
Supervised projects A54 A57 B30 B31 B32 B34 B35 B36 C14 Proposed projects. 15
 
Assessment comments

The students participating in the EHEA should take a minimum of 75% of the continuous assessment (c.a.) tests done in the classroom. These c.a. test might be posed either on keynote lectures or problem solving sessions, and they represent the 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.

All aspects related to "academic exemption", "dedication to study", "permanence" and "academic fraud" will be governed in accordance with the current academic regulations of the UDC. In any case, students with an academic dispensation are asked to indicate this to the teaching staff at the beginning of the semester.

Sources of information
Basic R.E. Larson, R.P. Hostetler, B.H. Edwards (1999). Cálculo. McGraw Hill
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 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
Villa, A. de la (). PROBLEMAS DE ALGEBRA LINEAL. GLAGSA


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

It is highly recommended to attend the introductory course that is held the first week of the first semester.


(*)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.