Degree in Physics

Objectives

The aim of this module is for students to develop skills in Linear Algebra, a branch of mathematics that is essential to the development of physics. Students are expected to understand matrix calculus from a conceptual point of view and to be able to apply it to solving problems specific to their field of study.

Prerequisites

There are no prerequisites

Competencies

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RK6 Understand the principles of mathematics and statistics that underpin the study of physics in classical and quantum systems

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RC3 Acquire IT knowledge and skills enabling the development of methods and technologies applicable to the relevant fields of study

Learning outcomes

RA10 Solves typical problems in linear algebra and applies the knowledge and skills acquired in relation to physics-related concepts.

RA11 Understands the abstract structure of vector spaces and is able to tackle problems of higher dimensionality.

RA12 Analyses, evaluates and interprets the results obtained when solving systems of equations.

LA13 Reason abstractly, using logical and algorithmic thinking, when solving exercises relating to linear applications and their matrix representations

RA14 Carries out work in the field of linear algebra through the knowledge and use of computer tools.

Description of the content

Topic 1: Matrices and systems of linear equations

Topic 2: Vector spaces

Topic 3: Homomorphisms between vector spaces

Topic 4: Diagonalisation of endomorphisms

Topic 5: Bilinear and quadratic forms. Scalar product. Norm.

Learning activities

AP1. Participatory lectures

AP2. Seminars or practical application sessions

AP3. Practical activities (case studies, project work, simulations, etc.)

AP4. – Independent study

AP5. Tutoring

AP6. Knowledge assessments

Assessment system and criteria

Without prejudice to any other requirements that may be specified in the relevant course syllabus, as a general rule, failure to attend more than 70 per cent of the course’s teaching activities—which require the student’s physical or virtual presence—will result in the loss of the right to continuous assessment during the standard examination period. In this case, the examination to be held during the official period set by the University shall be the sole assessment criterion, with the weighting specified in the course syllabus.

----

The assessment process will consist of evaluating the extent to which the student has acquired the competences associated with the module.

REGULAR EXAMINATION PERIOD – CONTINUOUS ASSESSMENT

VERY IMPORTANT: in order for students to benefit from continuous assessment, they must attend at least 70 per cent of the scheduled class hours (SESSION, WORKSHOP). If attendance is less than 70 per cent, the module may be passed by sitting a single examination during an official examination period (ordinary or supplementary). In the ordinary examination period, the weighting of this exam will be 60 per cent, in accordance with the ‘SE2 – Final Knowledge Tests’ assessment system, as set out in the report for the degree verification application, which means that a student who has forfeited their right to continuous assessment may obtain, at most, a mark of 6.0 out of 10 for the module in that sitting; to pass the module, they must achieve a mark of 8.34 out of 10 in the exam.

Continuous assessment will consist of the following components:

-- individual portfolio (for each student) SE4, accounting for 10 per cent of the final mark for the module, which will consist of evidence of the use of computer tools such as Wolfram Alpha, GeoGebra or any other (MATLAB, Mathematica, Python with NumPy, etc.) to solve various linear algebra problems throughout the term.

-- A mid-term exam (not a pass/fail exam) SE1, accounting for 30% of the final mark for the module. The date of this exam, which will take place during the term, will be announced well in advance; its content will cover topics 1 and 2.

-- A final exam (SE2), which will take place during the ordinary examination session (January), on the date set for that session by the university. This exam will assess all the content covered in the module, and it will account for 60 per cent of the final mark, provided that the student achieves a mark of 4.0 out of 10 or higher in the exam. Otherwise, the student’s final mark for the module will be the mark obtained in that examination.

***** Only the examinations will be subject to re-marking.

Should a student have forfeited their right to continuous assessment, they may, as described above, pass the module by sitting the examination covering all the content taught in the module during the ordinary examination session. However, as previously noted, this examination will account for 60 per cent of the final mark for the module; therefore, the student will need to achieve a minimum of 8.4 out of 10 in it in order to pass.

*** The module is considered to have been passed in the ordinary examination period if the final mark is 5.0 or higher.

SUPPLEMENTARY EXAMINATION PERIOD

Regardless of their attendance during the term, in the supplementary examination session students will be examined on all the content covered in the course in a single examination. The mark for this session will be that obtained in that examination (continuous assessment will not be taken into account).

*** The module is considered passed in the supplementary examination period if the final mark is 5.0 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

Bibliography

Essential:

1. David Lay

Linear Algebra and its Applications

PEARSON. 2016.

ISBN: 6073237456

2. Juan De Burgos Román

Algebra and Geometry: Definitions, Theorems and Results

García Maroto Editores. 2010.

ISBN: 9788492976942

3. Luis Merino and Evangelina Santos

Linear Algebra Using Elementary Methods

Paraninfo. 2010.

ISBN: 978-84-9732-4

4. Sheldon Axler

Linear Algebra Done Right

Springer. 2023.

ISBN: 3031410254

Supplementary:

5. Gilbert Strang

Introduction to Linear Algebra

Wellesley Cambridge Press. 2008.

ISBN: 8175968117

6. Juan De Burgos Román

Linear Algebra: 80 Useful Problems

García Maroto Publishers. 2007.

ISBN: 9788493601805

Others:

7. Eugenio Hernández

Linear Algebra and Geometry

3rd ed. ADDISON WESLEY. 2012.

ISBN: 9788478291298

8. Jesús Rojo

Linear Algebra

McGraw-Hill. 2001.

ISBN: 8448130162

9. Jesús Rojo

Exercises and Problems in Linear Algebra

2nd ed. McGraw-Hill. 2005.

ISBN: 8448198581

10. Stanley I. Grossman and José Job Flores

Linear Algebra

McGraw-Hill. 2012.

ISBN: 978-607-15-07

Objectives

The aim of this module, together with Analysis II, is to equip students with the necessary mathematical tools in the field of analysis to solve problems in the field of physics.

Prerequisites

No prerequisites

Learning Outcomes

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RK6 Understand the principles of mathematics and statistics that underpin the study of physics in classical and quantum systems

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RC3 Acquire IT knowledge and skills enabling the development of methods and technologies applicable to the relevant fields of study

Learning outcomes

RA1 Solve problems involving differential calculus of a real variable.

RA2 Develops and applies abstract reasoning and logical and algorithmic thinking when working with real functions of a real variable.

RA3 Performs operations with real numbers, developments involving numerical sequences and series, and mathematical calculations with mental and written agility.

LA4 Produces work in the field of differential and integral calculus of a single variable through the knowledge and use of computer tools.

Description of the content

1) Real numbers.

Sequences of real numbers: convergence.

Series of real numbers: convergence criteria.

Operations on series.

2) Real functions of a real variable.

Continuity.

The derivative: Rolle’s theorem, the mean value theorem, the generalised mean value theorem, L’Hopital’s

.

Applications of the derivative: Taylor’s formula with its error term, representation of functions.

3) Integration.

Riemann integral: the fundamental theorem of calculus.

Integration techniques: direct integration, integration by parts, substitution, rational, irrational and trigonometric integrals.

Applications: calculation of areas, lengths and volumes.

Teaching activities

AP1. – Participatory lectures 96 4 100

AP2. – Seminars or practical application classes 60 2.5 100

AP3.- Practical activities (case studies, project work, simulation, etc.) 144 3 50

AP4.- Independent study 240 0 0

AP5.- Tutoring 48 0.6 30

AP6. – Assessments 12 0.5 100

TOTAL 600 10.6

Assessment system and criteria

The mark for the resit will be calculated using the following methods:

SE1 – A mid-term test accounting for 30 per cent.

SE2 – The official exam in the standard assessment period, which will account for 60% of the mark and cover the entire course content. Under no circumstances will passing the mid-term test exempt students from this final exam.

SE4 – A portfolio (10%), comprising exercises set by the lecturer.

Continuous assessment (SE1 and SE4) will be forfeited in the following two circumstances:

1) if class attendance is less than or equal to 70 per cent

2) if the mark in the official examination is less than or equal to 4/10

Students who, for either of these reasons, fail the continuous assessment (SE1 and SE4) will still sit the official exam, which will account for 60 per cent of the mark, but the continuous assessment (SE1 and SE4) will be weighted at 0 out of 10.

For clarity, please note that even if the continuous assessment is not forfeited due to attendance, scoring less than 4/10 in the official examination will result in a fail for the module.

The resit exam will count for 100 per cent of the mark in this examination session.

Objectives

The aim of the module is to equip students with the necessary tools to tackle fundamental problems in the field of physics (historical context, kinematics, dynamics, energy and associated theorems, multi-particle systems), thereby providing them with the necessary foundation for subsequent modules.

Prerequisites

No prior requirements have been met.

Learning Outcomes

RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RS3 Estimate orders of magnitude to interpret laboratory phenomena in the field of Physics and its related disciplines, as well as in Chemistry.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RS5 Use appropriate electronic instruments and/or computer tools in modelling to find solutions to physics problems.

Learning outcomes

RA1 Identifies the relevant physical principles and, where necessary, makes simplifications and uses estimates of orders of magnitude in order to model and solve practical problems.

RA2 Handles fundamental concepts such as particles and fields, force, work, etc., with ease, in order to describe physical systems correctly.

LR3 Applies Newton’s laws appropriately to solving problems involving particles and systems of particles, and in relation to oscillatory motion.

RA4 Understands the units of the International System of Units and correctly assigns them to each of the physical quantities studied, as well as other units commonly used in the field of physics.

RC1 Work independently on the management of projects related to the different areas of physics

Description of the content

- Topic 1: Historical introduction.

- Topic 2: Particle kinematics and types of motion.

- Topic 3: Particle dynamics and the causes of motion.

- Topic 4: Work, energy and associated theorems.

- Topic 5: Systems of particles. Geometry of masses.

- Topic 6: Kinematics of a rigid body

- Topic 7: Dynamics of a rigid body

Learning activities

Learning activity No. of hours* Contact hours (8–12)** % contact time

AP1.- Participatory lectures 50 2.78 100

AP2.- Seminars or practical application classes 30 1.67 100

AP3.- Practical activities (case studies, project work, simulation, etc.) 72 2 50

AP4. – Independent study 180 0 0

AP5. – Tutorials 36 0.6 30

AP6.- Knowledge assessments 8 0.44 100

AP10.- Workshop and/or laboratory activities 74 4.11 100

TOTAL 450 11.60

Assessment system and criteria

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

ASSESSMENT SYSTEMS

The assessment systems for this module are:

- AS1: Practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.).

- AS2: Final knowledge assessments.

- AS3: Laboratory practical book

ASSESSMENT CRITERIA

The assessment methods described above are specified in the following assessment criteria. There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR EXAMINATION PERIOD+++

The final mark for this sitting is the weighted average of a set of assessment tasks detailed below:

- Practical activities (SE1) 30% of the final mark: A mid-term exam set in advance (20%) + Submission of exercises (10%).

- Final knowledge assessments (SE2) 50% of the final mark: Final exam for the ordinary assessment period, covering the entire course content.

- Laboratory practical logbook (SE3): 20 per cent of the final mark. Final report on the metrology practicals carried out in November.

For continuous assessment (comprising the Practical Activities (SE1) and the Laboratory Practical Logbook (SE3)) to be taken into account, students must achieve a mark of 4.0/10 or above in the final exam during the ordinary examination period. Otherwise, their mark will correspond directly to that obtained in that exam.

Furthermore, in order for students to benefit from continuous assessment, they must attend at least 70 per cent of the scheduled class hours (SESSION, LAB). If attendance is less than 70 per cent, the module may be passed by sitting a single examination during an official examination period (ordinary or supplementary). In the ordinary examination session, the weighting of this exam will be 60 per cent, in accordance with the ‘SE2 – Final Knowledge Tests’ assessment system, as set out in the report for the degree verification application, which means that a student who has forfeited their right to continuous assessment may obtain, at most, a mark of 6.0 out of 10 for the module in that sitting; to pass the module, they must achieve a mark of 8.34 out of 10 in the exam.

The module is considered to have been passed in the ordinary examination period if the mark obtained in accordance with the above guidelines is 5.0 or higher.

+++EXTRAORDINARY EXAMINATION PERIOD+++

If a student has not passed the module in the ordinary examination session, they may sit the extraordinary examination session.

The supplementary examination period will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

- 100 per cent of the final mark for the module in the supplementary examination session corresponds to the final exam for this session, which assesses the entire syllabus of the module.

The module is considered to have been passed in the extraordinary examination session if the final mark is 5.0 or higher.

GRADES

Article 5 of Royal Decree 1125/2003 of 5 September establishes the grading system applicable to modules within degree programmes falling within the scope of the European Higher Education Area. This system is as follows:

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative grade may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

The distinction of ‘Honours’ shall be awarded to students who have obtained a mark of 9.0 or higher. The number of students awarded this distinction may not exceed five per cent of those enrolled on the course in the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘First Class Honours’ may be awarded.

Timetable

Click on this link to view the detailed timetable in Excel

Objectives

The aim of the module is to provide students with the conceptual and methodological foundations for translating scientific problems into computational procedures, developing programmed solutions that enable them to explore, analyse and solve physical and mathematical models.

Prerequisites

None

Learning Outcomes

RC1 To work independently on the management of projects relating to different areas of physics

RC3 Acquire IT knowledge and skills enabling the development of methods and technologies applicable to the relevant fields of study

RODS Develop effective communication, teamwork, analytical thinking, creativity and ethical leadership from a cross-disciplinary perspective, drawing clearly on democratic principles and values as well as the Sustainable Development Goals, in order to conduct oneself with integrity in the professional sphere.

Learning outcomes

RA1 Describes the fundamentals of basic general-purpose programming structures.

RA2 Uses IT tools to solve physical problems and illustrate mathematical concepts.

LA3 Formulates algorithms and models simple physical problems in order to implement them in a programming language (Python, Java or similar).

LA4 Reads and analyses programmes in order to identify the problem they solve or to detect possible errors, as well as to make modifications to adapt them to solving other similar scientific problems.

Course description

- Introduction to programming languages. Fundamentals of programming. Fundamentals of Python, Java or similar.

- Application structure. Data and expressions.

- Control structures. Classes and objects.

- Libraries. Data types, data structures and data types. Data input/output.

- Fundamentals of computing. Computer architecture. Storage systems. Performance evaluation.

Training activities

Training activity No. of hours* Face-to-face hours (8–12)** % face-to-face

AP1.- Participatory lectures 48 4 100

AP2.- Seminars or practical application classes 30 2.5 100

AP3.- Practical activities (case studies, project work, simulations, etc.) 72 3 50

AP4.- Independent study 120 0 0

AP5. – Tutoring 24 0.6 30

AP6.- Knowledge assessments 6 0.5 100

TOTAL 300 10.6

Assessment system and criteria

Regular examination session

The final mark for this assessment period is the weighted average of the following components:

— Practical activities (SE1), 27.5 per cent: oral assessment (10 per cent), individual final project with classroom presentation (10 per cent) and group final project with classroom presentation (7.5 per cent).

— Final knowledge assessments (SE2), 60 per cent: final examination for the ordinary assessment period, which assesses the entire course content.

— Portfolio (SE4), 12.5 per cent: exercises completed and handed in during class (7.5 per cent) and UAX Skill School courses assigned to the module (5 per cent).

The course content is cumulative: no assessment is limited to a single topic.

Assessment criteria

— Oral assessment: understanding of one’s own code, mastery of the programme’s flow, justification of the chosen data types and structures, response to an impromptu modification of the question, and precision of terminology.

— Final assignments: functional correctness including edge cases, decomposition into functions, justified use of classes and objects, error handling via exceptions, coding style (PEP 8 in Python) and user documentation. In group work, the integration of the result and the Git history are also assessed. Submission entitles the student to the defence, and the mark is derived from this; failure to attend the defence results in the loss of 50 per cent of the mark for this activity.

— Final exam: reading and writing code, types and expressions, control structures, functions, classes, arrays, files, exceptions, and the fundamentals of computers and storage systems.

— Portfolio: completing tasks within the session time limit, correcting solutions, regularity of submissions and incorporation of corrections highlighted in previous exercises. UAX Skill School is assessed on the basis of the accredited completion of the assigned courses within the specified timeframe.

Minimum mark in the final exam (ordinary examination session)

In order for continuous assessment (Practical Activities (SE1) and Portfolio (SE4)) to be taken into account, students must achieve a mark of 5.0/10 or higher in the final exam of the standard examination session. Otherwise, their final mark will correspond directly to the mark obtained in that exam.

Attendance

In order for students to benefit from continuous assessment, they must attend at least 70 per cent of the scheduled class hours (SESSION and TRAB). Should attendance fall below 70 per cent, the module may be passed by sitting a single examination during an official sitting, whether ordinary or supplementary. In the ordinary examination session, the weighting of this exam will be 60 per cent, in accordance with the ‘SE2 – Final Knowledge Tests’ assessment system, as set out in the report for the degree verification application, which means that a student who has forfeited their right to continuous assessment may obtain, at most, a mark of 6.0 out of 10 for the module in that sitting; to pass the module, they must achieve a mark of 8.34 out of 10 in the examination.

Extraordinary examination session

The mark relates exclusively to the final examination in the extraordinary examination session. Continuous assessment activities do not count towards the mark, and no mark – whether full or partial – from the final examination in the ordinary examination session is retained. The module is passed if the mark for that examination is 5.0 or above.

Timetable

Click on this link to view the detailed timetable in Excel

Objectives

The aim of this module is to introduce students to the fundamental concepts of statistics and probability, with a focus on the analysis of data derived from physical phenomena and processes. Throughout the course, we will learn how to carry out statistical analysis of real-world data. Key topics such as descriptive statistics, probability calculations and statistical inference will be covered, with a particular emphasis on their application to physical problems.

The course also includes the use of software to solve real-world problems and analyse experimental data, enabling students to develop the skills needed to draw accurate and useful conclusions from the data. By the end of the course, students should be able to model, solve and analyse statistical problems using fundamental techniques and tools from data science. Prior knowledge of Algebra, Calculus and Programming will be helpful in following the course.

Prerequisites

No prerequisites have been set.

Learning Outcomes

RC1 Work independently on the management of projects related to the various areas of physics

RC3 Acquire IT knowledge and skills enabling the development of methods and technologies applicable to the relevant fields of study

RODS Develop effective communication, teamwork, analytical thinking, creativity and ethical leadership from a cross-disciplinary perspective, clearly inspired by democratic principles and values, as well as the Sustainable Development Goals, in order to conduct oneself with integrity in the professional sphere.

Learning outcomes

RA5 Apply basic knowledge of statistics and probability to the specific analysis of data relating to physical phenomena and processes.

RA6 Carries out statistical data analysis applied to the resolution of real-world cases using mathematical software development packages (Python + NumPy, SPSS or similar).

LA7 Applies probability calculations through the identification of models.

LA8 Describes and applies the basic principles of statistical inference.

Course content

The general content of the module relates to the fundamentals of descriptive statistics, counting techniques, distributions, partitions of a set and generating functions; probability theory, random variables and probability distributions; confidence intervals and statistical inference; and applications of probability and statistics to data analysis.

R will be used throughout the course as statistical software, alongside Python applications designed for use in data science.

This content will be organised into the following topics:

Topic 1. Descriptive statistics.

Topic 2. Probability.

Topic 3. Random variables.

Topic 4. Probability distributions.

Topic 5. Statistical inference.

Topic 6. Linear regression.

Topic 7. Statistics applied to data analysis.

Teaching activities

AP1. – Participatory lectures 48 4 100

AP2. Seminars or practical application sessions 30 2.5 100

AP3. Practical activities (case studies, project work, simulations, etc.) 72 3 50

AP4. Independent study 120 0 0

AP5.- Tutoring 24 0.6 30

AP6.- Knowledge assessments 6 0.5 100

Assessment system and criteria

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

ASSESSMENT SYSTEMS

The assessment systems for this module are:

- AS1: Practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.).

- AS2: Final knowledge assessments.

- AS4: Portfolio.

ASSESSMENT CRITERIA

The assessment methods described above are set out in the following assessment criteria. There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR SESSION+++

The final mark for this sitting is the weighted average of a set of assessment tests detailed below:

- Practical activities (SE1) 30% of the final mark: Statistical project using software (30%)

- Portfolio (SE4) 10% of the final mark: Submission of exercises set to be completed in class.

- Final knowledge assessments (SE2) 60% of the final mark: Final exam for the ordinary assessment period, covering the entire course content.

For continuous assessment (comprising the Practical activities (SE1) and Portfolio (SE4)) to be taken into account, students must achieve a mark of 4.0/10 or higher in the final exam during the regular examination period. (SE2) Otherwise, their mark will correspond directly to that obtained in that exam.

Furthermore, in order for students to benefit from continuous assessment, they must attend at least 70 per cent of the scheduled class hours (SESSION, WORKSHOP). If attendance is less than 70 per cent, the module may be passed by sitting a single examination during an official examination period (regular or supplementary). In the ordinary examination session, the weighting of this examination will be 60 per cent, in accordance with the ‘SE2 – Final Knowledge Tests’ assessment system set out in the report for the application for degree accreditation, which means that a student who has forfeited their right to continuous assessment may obtain, at most, a mark of 6.0 out of 10 for the module in that sitting; to pass the module, they must therefore achieve a mark of 8.34 out of 10 in the examination.

The module is considered to have been passed in the ordinary examination period if the mark obtained in accordance with the above guidelines is 5.0 or higher.

+++EXTRAORDINARY EXAMINATION PERIOD+++

If a student has not passed the module in the ordinary examination session, they may sit the extraordinary examination session.

The supplementary examination period will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

- 100 per cent of the final mark for the module in the supplementary examination period is based on the final exam for this period, which assesses the entire syllabus of the module.

The module is considered to have been passed in the extraordinary examination period if the final mark is 5.0 or higher.

GRADES

Article 5 of Royal Decree 1125/2003 of 5 September establishes the grading system applicable to modules within degree programmes falling within the scope of the European Higher Education Area. This system is as follows:

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative grade may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

The distinction of ‘Honours’ shall be awarded to students who have obtained a mark of 9.0 or higher. The number of students awarded this distinction may not exceed five per cent of those enrolled on the course in the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘First Class Honours’ may be awarded.

Timetable

Click on this link to view the detailed timetable in Excel

Bibliography

Essential:

1. Bruce, Peter

Practical Statistics for Data Science with R and Python/

2nd ed. Marcombo, 2022.

ISBN: 9788426734433

2. Navidi, William.

Statistics for Engineers and Scientists/

5th ed. McGraw-Hill, 2022.

ISBN: 9781456293147

Objectives

This module, together with Algebra I, aims not only to familiarise students with the main results of linear algebra, but also to enable them to understand matrix calculus from a conceptual point of view and to apply it to solving problems typical of physics; it therefore forms part of the mathematical foundation not only for other subjects within the degree programme, but also essential for the professional practice of a physics graduate.

Algebra II will cover what is known as linear geometry, which encompasses the study of vector spaces and Euclidean affine spaces, affine varieties, affine mappings and transformations (in particular, rigid transformations), the algebraic description of conics and quadric surfaces and, finally, an introduction to projective geometry.

Throughout this module, students will continue to be trained in the use of computer tools for solving problems in linear algebra and, in particular, in the associated geometry.

Prerequisites

No prerequisites

Learning outcomes

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RK6 Understand the principles of mathematics and statistics that underpin the study of physics in classical and quantum systems

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RC3 Acquire IT knowledge and skills enabling the development of methods and technologies applicable to the relevant fields of study

Learning outcomes

RA15 Solves typical problems in linear geometry and applies the knowledge and skills acquired in relation to physics-related concepts.

RA16 Demonstrates the ability to distinguish between quadratic and conic sections.

RA17 Analyses, evaluates and interprets the results obtained when solving problems in linear geometry.

LA18 Explains affine transformations and movements in an affine Euclidean space and solves exercises related to these concepts.

RA19 Produces work in the field of linear geometry by applying knowledge and using computer-based tools.

Description of the content

- Euclidean vector spaces.

- Affine spaces. Linear varieties: lines and planes. Affine transformations.

- Euclidean affine spaces. Transformations in a Euclidean affine space.

- Spectral theory of symmetric and orthogonal matrices for application in the study of transformations.

- Conics and quadric surfaces.

- Projective geometry.

Teaching activities

AP1. – Interactive lectures

AP2. Seminars or practical application sessions

AP3.- Practical activities (case studies, project work, simulations, etc.)

AP4. Independent study

AP5. Tutoring

Assessment system and criteria

The assessment process will consist of evaluating the extent to which the student has acquired the competences associated with the module.

REGULAR EXAMINATION PERIOD – CONTINUOUS ASSESSMENT

VERY IMPORTANT: in order for students to benefit from continuous assessment, they must attend at least 70 per cent of the scheduled class sessions (SESSION, WORK). If attendance is less than 70 per cent, the module must be passed by sitting a single examination during an official examination period (ordinary or supplementary). In the ordinary examination period, the weighting of this exam will be 60 per cent, in accordance with the ‘SE2 – Final Knowledge Tests’ assessment system, as set out in the report for the degree verification application, which means that a student who has forfeited their right to continuous assessment due to attendance below 70 per cent may obtain, at most, a mark of 6.0 points for the module in that examination session; to pass the module, they must therefore achieve a mark of 8.34 points in the examination.

Continuous assessment will consist of the following components:

-- portfolio (SE4), individual, accounting for 10 per cent of the final mark for the module, which will consist of evidence of the use of computer tools such as Wolfram Alpha, GeoGebra or any other (MATLAB, Mathematica, Python with NumPy, etc.) to solve various linear algebra problems throughout the term.

-- a practical case study (SE1), accounting for 10 per cent of the final course mark, which students will undertake in small groups throughout the academic term; this practical case study will involve the submission of several deliverables (in order to analyse the progress of the different working groups), each of which will be assessed and will carry its corresponding weighting in the case study’s mark. Submission dates will be announced well in advance.

-- a mid-term exam (SE1), accounting for 20 per cent of the final mark for the module. The date of this exam, which will take place during the term, will be announced well in advance; it will cover topics 1 and 2.

-- a final examination (SE2) accounting for 60 per cent of the final mark for the module. This will take place during the regular examination period (January), on the date set by the university for that period. A minimum mark of 4/10 is required to pass the module.

*** The module is deemed to have been passed in the ordinary examination period if the final mark is 5.0 or higher.

SUPPLEMENTARY EXAMINATION PERIOD

Regardless of their attendance during the term, in the extraordinary examination period, students will be examined on all the content covered in the module in a single examination. The mark for this examination period will be the mark obtained in that examination (continuous assessment will not be taken into account).

*** The module is considered passed in the extraordinary sitting if the final mark is 5.0 or higher.

Bibliography

Core:

1. Juan De Burgos Román

Algebra and Geometry. Definitions, Theorems and Results

García Maroto Editores. 2010.

ISBN: 9788492976942

2. Luis Merino and Evangelina Santos

Linear Algebra using Elementary Methods

Paraninfo. 2010.

ISBN: 978-84-9732-4

Supplementary:

3. Gilbert Strang

Introduction to Linear Algebra

Wellesley Cambridge Press. 2008.

ISBN: 8175968117

4.- Juan De Burgos Román

Linear Algebra. 80 Useful Problems

García Maroto Editores. 2007.

ISBN: 9788493601805

Others:

5. Eugenio Hernández

Linear Algebra and Geometry

3rd ed. ADDISON WESLEY. 2012.

ISBN: 9788478291298

6. Jesús Rojo

Linear Algebra

McGraw-Hill. 2001.

ISBN: 8448130162

7. Jesús Rojo

Exercises and Problems in Linear Algebra

2nd ed. McGraw-Hill. 2005.

ISBN: 8448198581

8. Stanley I. Grossman and José Job Flores

Linear Algebra

McGraw-Hill. 2012.

ISBN: 978-607-15-07

Objectives

The aim of this module is to provide students with a solid grounding in differential and integral calculus of several variables, enabling them to tackle complex mathematical problems applied to physical phenomena. By the end of the module, students will be able to:

Understand and analyse the geometry of Euclidean space, applying concepts of vectors, planes and surfaces to the solution of physical problems.

Study functions of several variables, examining their properties relating to limits, continuity and differentiability.

Calculate higher-order derivatives and apply Taylor’s theorem in several variables to approximate functions and solve optimisation problems.

Work with vector functions, analysing trajectories and curves in space, as well as their properties of continuity and differentiability.

Calculate multiple integrals (double and triple) by applying Fubini’s theorem and the change of variables theorem, with applications to the calculation of volumes and masses.

Solve line integrals and surface integrals, applying them to the study of vector fields and flux through surfaces.

Apply the fundamental theorems of vector analysis (Green’s, Gauss’s and Stokes’s) to physics problems, such as electrostatics and fluid dynamics.

Interpret and analyse the results obtained from mathematical calculations, applying them to the modelling and solution of physical problems.

Prerequisites

No prerequisites

Competencies

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RK6 Understand the principles of mathematics and statistics that underpin the study of physics in classical and quantum systems

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RC3 Acquire IT knowledge and skills enabling the development of methods and technologies applicable to the relevant fields of study

Learning outcomes

RA5 Applies integrals in calculations associated with physics-related concepts.

RA6 Explain the geometry of multidimensional space and demonstrate three-dimensional spatial awareness.

RA7 Uses specific formal mathematical language with rigour

RA8 Produces work in the field of integral calculus through the knowledge and use of computer tools.

RA9 Analyses, evaluates and interprets the results obtained when solving problems involving differential calculus of several real variables.

Description of the content

Topic 1. The geometry of Euclidean space

Vectors in Euclidean space: magnitude, direction and vector operations.

Planes and surfaces: equations and geometric properties.

Topic 2. Functions of several variables: limits and continuity. Differentiation.

Functions of several variables: domain, range and graphs.

Limits and continuity in several variables.

Partial derivatives and differentiability.

Tangent plane and linear approximation.

Topic 3. Higher-order derivatives

Higher-order partial derivatives.

Hessian and convexity criteria.

Taylor’s theorem in several variables.

Applications in optimisation and stability analysis.

Topic 4. Functions with vector-valued outputs

Vector functions: continuity, differentiability and applications.

Parametric curves in space.

Derivatives of vector functions: velocity and acceleration.

Vector fields and lines of flow.

Topic 5. Double and triple integrals. Fubini’s theorem and the change of variables

Double and triple integrals: definition and properties.

Change of order of integration: Fubini’s theorem.

Change of variables in multiple integrals.

Applications to the calculation of areas, volumes, masses and centres of mass.

Topic 6. Integrals over curves and surfaces

Line integrals: over scalar and vector fields.

Surface integrals: flux through a surface.

Topic 7. Integration theorems in vector calculus

Green’s theorem: circulation and flux in the plane.

Gauss’s theorem (divergence): flux through closed surfaces.

Stokes’ theorem: relationship between circulation and rotational flux.

Learning activities

AP1. Participatory lectures

AP2. Seminars or practical application classes

AP3. Practical activities (case study sessions, project work, simulation, etc.)

AP4. Independent study

AP5. Tutoring

Assessment system and criteria

Regular examination period

VERY IMPORTANT: in order for students to benefit from continuous assessment during the ordinary assessment period, they must attend at least 70 per cent of the scheduled class sessions (SESSION, WORK).

Practical group activities (20 per cent): Solving physics-related problems using analytical tools.

Submission of exercises (20 per cent): Individual exercises to consolidate theoretical and practical concepts.

Final exam (60 per cent): A minimum mark of 4 is required to be included in the average.

Supplementary examination

The final mark for the module will correspond to 100 per cent of the mark obtained in this exam.

Bibliography

Core:

1. JERROLD E. MARSDEN and ANTHONY J. TROMBA

Vector Calculus.

Pearson. 2004.

ISBN: 9780201629354

Objectives

To provide students with the physical knowledge of fluid mechanics, electrical phenomena and magnetic fields necessary to tackle more complex physical problems.

Prerequisites

No prerequisites have been specified.

Learning Outcomes

RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RS3 Estimate orders of magnitude to interpret laboratory phenomena in the field of physics and its related disciplines, as well as in chemistry.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RS5 Use appropriate electronic instruments and/or computer tools in modelling to find solutions to physics problems.

RC1 Work independently on the management of projects related to the different areas of physics

Learning outcomes

RA5 Understands, at a basic level, the shortcomings of so-called Classical Physics and of the experiments that led to the formulation of Special Relativity.

RA6 Applies the knowledge acquired to formulate and solve problems, identifying the relevant physical principles and using order-of-magnitude estimates in relation to fluids, thermodynamics, waves, electric fields and magnetic fields.

LA7 Interprets electrical and magnetic phenomena in nature in terms of electromagnetic fields and their interactions with matter.

RA8 Develops a broad overview of the scope of modern physics.

Course content

- Rigid bodies

- Fluids (statics and dynamics)

- Wave motion.

- Electric field.

- Magnetic field.

- Introduction to thermodynamics.

- Optics.

- Special relativity and modern physics.

Teaching activities

AP1. – Participatory lectures

AP2. Seminars or practical application classes

AP3. Practical activities (case studies, project work, simulations, etc.)

AP4. – Independent study

AP5. Tutoring

AP6. Assessment tests

AP10.- Workshop and/or laboratory activities

Assessment system and criteria

Regular examination period

VERY IMPORTANT: in order for students to benefit from continuous assessment during the ordinary assessment period, they must attend at least 70 per cent of scheduled class hours (SESSION, LAB).

Assessment system Weighting

SE1.- Practical activities (case studies, problem-solving and challenges, possible project to boost marks)  20 %

SE2. Final knowledge assessments  (with a minimum mark of 4.5/10 to be included in the average with the other assessment components) 50 %

SE3. Mid-term knowledge assessments (with a minimum mark of 4.5/10 to be included in the average with the other assessment methods) 30 per cent

Extraordinary examination session

The final mark for the module will correspond to 100 per cent of the mark obtained in this examination

Objectives

The objectives set out for the ‘Fundamentals of Chemistry’ module within the Bachelor’s Degree in Physics uphold effective equality between women and men as established by Organic Law 3/2207 of 22 March, the principles of equal opportunities, non-discrimination and universal accessibility for people with disabilities, as set out in Act 51/2203 of 2 December, and promote education for peace, non-violence and human rights, as set out in Act 27/2005 of 30 November.

The specific objectives proposed for the ‘Fundamentals of Chemistry’ module are as follows:

To provide training in the sciences, with a particular focus on chemistry, enabling students to undertake the study of technological subjects.

To ensure the acquisition of cross-curricular competences and skills that enable and enhance the application of the knowledge acquired.

To foster the capacity for innovation and the dissemination of scientific findings.

To foster a commitment to ethical standards in both professional and social contexts.

Prerequisites

No prerequisites.

Competencies

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of Physics.

RS3 Estimate orders of magnitude to interpret laboratory phenomena in the field of physics and its sub-disciplines, as well as in chemistry.

Learning outcomes

RA1 List, at a basic level, the principles that explain the physicochemical properties of matter.

RA2 Formulate and name simple inorganic compounds.

RA3 Describe the main mechanisms involved in a chemical reaction, identify different types of reaction, determine the quantities of reactants, and use thermodynamic potentials to characterise the reaction in terms of energy.

LA4 Determine the mechanisms responsible for chemical equilibrium and explain how the parameters on which it depends act.

RA5 Determines the acidity of a solution.

RA6 Identifies some of the main organic functional groups and determines their most important chemical reactions.

RA7 Analyses, evaluates and interprets the results obtained when solving problems.

Description of the content

- Atoms and chemical bonding. Inorganic formulas and nomenclature. Intermolecular forces.

- States of matter and phase diagrams. Fundamentals of thermochemistry. Mixtures. Solutions.

- Chemical reactions: mechanisms and rates; stoichiometry.

- Chemical equilibrium: constants; Le Chatelier’s principle. Solubility equilibrium. Acids and bases.

- Oxidation-reduction reactions and electrochemistry. Introduction to organic chemistry.

- Oxidation-reduction reactions and electrochemistry. Introduction to organic chemistry.

Teaching activities

AP1.- Participatory lectures

AP2. – Seminars or practical application sessions

AP3.- Practical activities (case study sessions, project work, simulations, etc.)

AP4. – Independent study

AP5. Tutoring

AP6. Assessment tests

AP10.- Workshop and/or laboratory activities

Assessment system and criteria

REGULAR EXAM SESSION

Both parts, Chemistry and Physics: Same assessment system. The weighted average of both parts must be 5 (with a minimum mark of 4 in each part to be included in the average) to pass.

IT WILL NOT BE POSSIBLE TO PASS THE SUBJECT BASED ON MID-TERM EXAMS.

A) CONTINUOUS ASSESSMENT with group assignments

- 15% practical work. Minimum mark of 5.0 to pass.

- 10% Mid-term 1, treated as a group assignment

- 5% class activities (group assignments on the virtual campus)

- 5% inorganic chemistry problem-solving exam

- 5% organic chemistry problem-solving exam

B) FINAL EXAM - Two parts (Chemistry and Physics)

- Students must sit the standard examination for both the Chemistry and Physics sections.

- Should a student have failed the laboratory examination for either the inorganic or organic chemistry modules, they may also retake this in the final examination.

- The mark obtained in the course activities will be retained when calculating the final mark.

SPECIAL EXAMINATION SESSION

In the final exam for the supplementary sitting, the entire module (Chemistry, Physics and Practical) will be assessed, and the mark for this exam will account for 100 per cent of the final mark for the module.

Objectives

To carry out laboratory measurements in accordance with established protocols, involving the calibration of instruments, the estimation of systematic and random uncertainties and the identification of strategies to eliminate them, the collection of data and its mathematical analysis.

Establish measurement protocols, particularly those relating to the safety of the experimenter.

To prepare reports on measurement procedures, the analysis of results and the conclusions drawn.

Prerequisites

No prior requirements.

Competencies

RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

RK3 Analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RS3 Estimate orders of magnitude to interpret laboratory phenomena in the field of physics and its related disciplines, as well as in chemistry.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RS5 Use appropriate electronic instruments and/or computer tools in modelling to find solutions to physics problems.

RC1 Work independently on the management of projects related to the different areas of physics

Learning outcomes

RA9 Carries out laboratory measurements in accordance with established protocols, including the calibration of instruments, the estimation of systematic and random uncertainties (identifying strategies for their elimination), the collection of data and the mathematical analysis of that data.

LA10 Follow measurement protocols, particularly those relating to the safety of the experimenter.

RA11 Prepares reports on measurement procedures, the analysis of results and the conclusions drawn.

Description of the content

- The nature of physical phenomena and their measurement.

- Processing of experimental data and calculation of errors.

- General physics laboratory practicals related to Fundamentals of Physics I and Fundamentals of Physics II

Teaching activities

AP1.- Participatory lectures

AP2.- Seminars or practical application sessions

AP3.- Practical activities (case studies, project work, simulations, etc.)

AP4. Independent study

AP5. Tutoring

AP6. Assessment tests

AP10.- Workshop and/or laboratory activities

Assessment system and criteria

Regular examination period

VERY IMPORTANT: in order for students to benefit from continuous assessment during the ordinary assessment period, they must attend at least 70 per cent of the scheduled class hours (SESSION, LAB).

Assessment system Weighting

SE1.- Practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.)  10

SE2. Final knowledge assessments (with a minimum mark of 4/10 to be included in the average with the other assessment components) 50

SE3. Laboratory practical workbook (students must submit the provided scripts for the practicals carried out for assessment) 40

Extraordinary examination session

The final mark for the module will correspond to 100 per cent of the mark obtained in this examination

Objectives

To understand and apply methods for solving ordinary differential equations and linear and non-linear systems. To develop analytical skills through transforms, series and dynamic models. To encourage the practical application of theory through problems, projects and exercises.

Prerequisites

None

Competencies

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RK6 Understand the principles of mathematics and statistics that underpin the study of physics in classical and quantum systems

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

Learning outcomes

LA1 Solves ordinary differential equations, linear differential equations and systems of non-linear equations.

RA2 Demonstrates proficiency in applying solution methods to basic physics equations.

RA3 Analyses the results and interprets the solutions to differential equations.

LA4 Carries out work by understanding, selecting and applying the operational method best suited to a particular problem.

Course content

- Methods for solving Ordinary Differential Equations (ODEs): Basic concepts and classification of ODEs. Equations with separable variables, exact equations and the integrating factor.

- First-order differential equations and higher-order linear equations.

- Systems of first-order differential equations: homogeneous and non-homogeneous linear systems.

- Non-linear differential equations and systems. Stability theory.

- Laplace transform and its applications to differential equations.

Learning activities

Learning activity No. of hours* Contact hours

(8–12)** % Face-to-face

AP1.- Participatory lectures 24 4 100

AP2. – Seminars or practical application sessions 15 2.5 100

AP3.- Practical activities (case studies, project work, simulations, etc.) 36 3 50

AP4. – Independent study 60 0 0

AP5.- Tutorials 12 0.6 30

AP6.- Knowledge assessments 3 0.5 100

TOTAL 150 10.6

Assessment system and criteria

ASSESSMENT CRITERIA

Assessment of the module will be carried out on a continuous basis and through a final assessment, combining practical activities, knowledge tests and the creation of a portfolio. The aim is thus to assess both the acquisition of theoretical skills and the practical application of the course content, as well as the student’s individual progress throughout the course. There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR EXAMINATION PERIOD+++

The final mark for this sitting is the weighted average of a set of assessment tasks detailed below:

- Practical activities (SE1) 30% of the final mark: A mid-term exam set in advance (20%) + Assignments set by the lecturer (10%).

- Final knowledge assessments (SE2) 60% of the final mark:

A written examination designed to assess students’ understanding of fundamental theoretical concepts and their ability to apply methods for solving differential equations in different contexts. Mid-term assessments do not count towards the final mark; therefore, the final exam will cover the entire syllabus of the course. The exam must, without exception, be taken on the date set by the University for the standard examination period (virtual campus).

- Portfolio (SE4) 10% of the final mark. This assessment system involves the daily submission of questions and problems not solved in class by the lecturer. Evidence of the student’s learning, compiled in a dossier, folder or portfolio throughout the course, is assessed with the aim of explaining and reflecting on the learning that has taken place.

For the continuous assessment (comprising the Practical Activities (SE1) and the Portfolio (SE4)) to be taken into account, students must achieve a mark of 4.0/10 or higher in the final exam of the standard examination session. Otherwise, their final mark will correspond directly to the mark obtained in that exam.

The module is considered passed in the ordinary examination period if the mark obtained in accordance with the above guidelines is 5.0 or higher.

+++SUPPLEMENTARY EXAMINATION PERIOD+++

If a student has not passed the module in the ordinary examination period, they may sit the extraordinary examination.

The supplementary examination period will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

- 100 per cent of the final mark for the module in the extraordinary sitting corresponds to the final exam for this sitting, which assesses the entire syllabus of the module.

The module is considered to have been passed in the extraordinary examination period if the final mark is 5.0 or higher.

Bibliography

Essential:

1. Dennis G. Zill

Differential Equations with Modelling Applications

Cengage Learning. 2001.

ISBN: 60-7526-631-3

2. George F. Simmons

Differential Equations: With Applications and Historical Notes

McGraw-Hill. 1993.

ISBN: 84-4810-045-X

Objectives

Students will learn the fundamental concepts of electromagnetism, with particular emphasis on the effective application of vector calculus to relevant problems in electrostatics, magnetostatics and problems involving boundary conditions.

Prerequisites

None

Learning Outcomes

RA1 Has a good grasp of the basic principles governing the generation of electromagnetic fields by charges and currents, and the effect of these fields on charges.

RA2 Understands how material media behave in the presence of electric and magnetic fields and knows how to calculate these fields.

RA3 Understands and applies analytical and numerical techniques relating to boundary value problems for the potential.

RA4 Understands and knows how to use Maxwell’s equations in their differential and integral forms.

Learning outcomes

o RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

o RK2 Understand physically distinct phenomena and their underlying analogies, enabling the application of known solutions to new problems

o RK3 Analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

o RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena they describe

o RK11 Understand the fundamentals and basic concepts of electric and magnetic fields, as well as the interrelationship between these fields and their unification in electromagnetism

o RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

o RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

Course content

-Topic 1. Electrostatics

The electric field (discrete and continuous charge distributions), divergence and curl of the electric field (Gauss’s law and irrotationality of electrostatic fields), Electric potential (Poisson’s equation, Laplace’s equation and boundary conditions), work and energy in electrostatics

-Topic 2. Conductive media

Basic properties, Induced charges, Surface charge and force on a conductor, Capacitors

-Topic 3. Magnetostatics and Electric Current

Magnetic forces (Lorentz forces and steady-state currents), Biot–Savart law, divergence and curl of the magnetic field (Ampère’s law, non-divergence of the magnetic field)

Training activities

Learning activity | No. of hours (8–12) | Contact hours | % Contact time

AP1.- Participatory lectures | 48 | 4| 100

AP2.- Seminars or practical application classes |30 |2.5 |100

AP3.- Practical activities (case studies, project work, simulations, etc.) |72 |3 |50

AP4.- Independent study |120 |0 |0

AP5. – Tutorials | 24 |0.6 |30

AP6. – Knowledge assessments |6 |0.5 |100

TOTAL 300 10.6

Assessment system and criteria

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

ASSESSMENT SYSTEMS

The assessment systems for this module are:

- AS1: Practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.).

- AS2: Final knowledge assessments.

- AS4: Portfolio.

ASSESSMENT CRITERIA

The assessment methods described above are set out in the following assessment criteria. There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR SESSION+++

The final mark for this sitting is the weighted average of a set of assessment tasks detailed below:

- Practical activities (SE1) 30% of the final mark: A mid-term exam set in advance (20%) + Assignments set by the lecturer (10%).

- Final knowledge assessments (SE2) 60% of the final mark: Final exam for the ordinary assessment period, covering the entire course content.

- Portfolio (SE4) 10% of the final mark. This assessment system involves the daily submission of questions and problems not solved in class by the lecturer. This assessment system will be weighted against the student’s overall attendance for the course: if, at the end of the academic year, the student’s attendance at the scheduled class sessions (SESSION, LAB) is 60 per cent or more, the mark for this system will be that obtained following the lecturer’s marking. If the student’s attendance at all scheduled course sessions (SESSION and LAB) is less than 60 per cent, the mark for this assessment system, as determined by the lecturer, will be reduced in proportion to the level of attendance achieved, in accordance with an attendance criterion defined by the lecturer.

In order for the continuous assessment (comprising the Practical Activities (SE1) and the portfolio (SE4)) to be taken into account, students must achieve a mark of 4.0/10 or higher in the final exam of the ordinary examination session. Otherwise, their mark will correspond directly to that obtained in that exam.

The module is considered passed in the standard examination period if the mark obtained in accordance with the above guidelines is 5.0 or higher.

+++SUPPLEMENTARY EXAMINATION PERIOD+++

If a student has not passed the module in the ordinary examination period, they may sit the extraordinary examination.

The supplementary examination session will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

- 100 per cent of the final mark for the module in the supplementary examination session corresponds to the final exam for this session, which assesses the entire syllabus of the module.

The module is considered to have been passed in the extraordinary examination period if the final mark is 5.0 or higher.

GRADES

Article 5 of Royal Decree 1125/2003, of 5 September, establishes the grading system applicable to modules in degree programmes within the European Higher Education Area. This system is as follows:

The award of the corresponding credits is contingent upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as a numerical mark on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative grade may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

The distinction of ‘Honours’ shall be awarded to students who have obtained a mark of 9.0 or higher. The number of students awarded this distinction may not exceed five per cent of those enrolled on the course in the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘First Class Honours’ may be awarded.

Bibliography

Essential:

1. Griffiths, David J

Introduction to Electrodynamics / David J. Griffiths

Harlow, UK: Pearson. 2014.

ISBN: 1108420419

2. Richard P. Feynman, Robert B. Leighton, Matthew Sands

The Feynman Lectures on Physics, Vol. II The New Millennium Edition: Mainly Electromagnetism and Matter

Basic Books. 2011.

ISBN: 9780465024940

Objectives

Technical English is a four-level course aimed at students on technical or vocational courses, as well as employees undergoing on-the-job training. It covers the basic language and skills required to communicate effectively in various technical fields.

Course description

The selected material presents technical content in a clear, structured and accessible manner, combining motivational texts with practical examples and illustrations that facilitate the understanding of concepts specific to the field of engineering.

The content is geared towards current situations and technological developments, aiming to meet students’ communication needs and encourage the use of English in real technical and professional contexts.

Assessment system and criteria

Continuous assessment

• Two written tests covering Units 1–8. Each will assess:

o Listening

o Reading comprehension (Reading)

o Grammar

o Vocabulary and subject-specific terminology (Terminology)

• One oral test, held on a separate date.

Weighting

• Written tests (2): 70%

• Oral examination: 20%

• Assessed activities: 10%

Addendum

ADDENDUM

Assessment tests: Students must sit all the assessment tests scheduled during the continuous assessment period. Failure to sit any of the mid-term assessments will result in the loss of the right to continuous assessment, and the student will be required to sit the Ordinary Examination, in which 100 per cent of the course content will be assessed.

Oral presentation: The topic of the oral presentation, which may be delivered individually or in pairs, must be agreed in advance with the lecturer.

Examination schedule: The dates for the various assessment tests will be announced well in advance. Unless otherwise specified, the tests will take place in the course’s usual classroom.

Textbook

Technical English 4, Second Edition

Pearson

Author: David Bonamy

Other books: 1. Scientific English: A Guide for Scientists and Other Professionals

Authors: Robert A. Day & Nancy Sakaduski

2. The Scientist’s Guide to Writing (2nd Edition)

Timetable

Click on this link to view the detailed timetable in Excel

Bibliography

Core:

1. Christopher Jacques

Technical English, 2nd Edition. Level 4. Workbook (with Answer Key and Audio CD Pack)

Pearson. 2022.

ISBN: 1292424532

2. David Bonamy

Technical English, 2nd Edition. Level 4. Coursebook and eBook

Pearson. 2022.

ISBN: 1292424494

Supplementary:

3.- Cambridge English for Scientists

Tamzen Armer

Cambridge. 2011.

ISBN: 3125351863

Objectives

This module, which together with Mechanics and Waves II forms the subject of Mechanics and Waves, has as its main objective to familiarise students with the Newtonian formulation of classical mechanics and to enable them to apply it correctly to the solution of mechanical problems. More specifically, students must understand the basic concepts of oscillatory motion, as well as the associated phenomena; they must be able to describe both the kinematics and dynamics of a rigid body in a plane and in space; and they must understand the relationship between symmetries and conservation laws in physics.

Prerequisites

None

Competencies

RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RK5 To understand the scope and limitations of classical physics that led to the formulation of special and general relativity, as well as quantum mechanics, in order to address the new problems arising in modern physics.

RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena described by them

RK12 Understand the theories, laws and models governing physical phenomena related to mechanics

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

Learning outcomes

RA1 Applies the Newtonian formulation appropriately to the solution of mechanical problems.

RA2 Understands the basic concepts of wave motion, as well as the basic phenomenology of oscillatory motion, including coupled oscillations and resonance, and is able to solve problems involving free, forced and damped oscillations.

LA3 Describes both the kinematics and dynamics of a rigid body in a plane and in space, and applies this knowledge to solving problems involving rigid bodies.

RA4 Relates symmetries and conservation laws in physics, applying them to the solution of practical exercises.

Course content

The module consists of the following topics:

1. Particle kinematics.

2. Newtonian dynamics.

3. Non-inertial reference frames.

4. Kinematics of rigid bodies.

5. Dynamics of a rigid body.

6. Oscillations and waves.

Teaching activities

Learning activity No. of hours* Contact hours

(8–12)** % Face-to-face

AP1. – Participatory lectures 48 4 100

AP2.- Seminars or practical application classes 30 2.5 100

AP3.- Practical activities (case studies, project work, simulations, etc.) 72 3 50

AP4. – Independent study 120 0 0

AP5.- Tutorials 24 0.6 30

AP6.- Knowledge assessments 6 0.5 100

TOTAL 300 10.6

Assessment system and criteria

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

ASSESSMENT SYSTEMS

The assessment methods for this module are:

- AS1: Practical activities (case studies, problem-solving and challenges, project work, oral presentations, debates, etc.).

- AS1: Final knowledge assessments.

- AS4: Portfolio.

ASSESSMENT CRITERIA

The assessment methods described above are set out in the following assessment criteria. There are two official examination sessions: the ordinary and the supplementary.

+++REGULAR SESSION+++

The final mark for this sitting is the weighted average of a set of assessment tasks detailed below:

- Practical activities (SE1) 30% of the final mark: Mid-term exam I (15%) + Mid-term exam II (15%).

- Final knowledge assessments (SE2) 60% of the final mark: Final exam for the ordinary examination period, covering the entire syllabus.

- Portfolio (SE4): 10% of the final mark. This assessment system involves completing and submitting exercises carried out in class, with non-trivial modifications designed to encourage creativity.

For continuous assessment (comprising the Practical Activities (SE1) and the portfolio (SE4)) to be taken into account, students must achieve a mark of 4.0/10 or higher in the final exam during the ordinary examination period. Otherwise, their mark will correspond directly to that obtained in the final exam.

The module is considered passed in the ordinary examination period if the mark obtained in accordance with the above guidelines is 5.0 or higher.

+++SUPPLEMENTARY EXAMINATION PERIOD+++

If a student has not passed the module during the ordinary examination period, they may sit the extraordinary examination.

The supplementary examination period will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

100% of the final mark for the supplementary sitting: Final examination for the supplementary sitting, which assesses the entire course content.

The module is considered passed in the extraordinary sitting if the final mark is 5.0 or above.

GRADES

Article 5 of Royal Decree 1125/2003 of 5 September establishes the grading system applicable to modules in degree programmes within the European Higher Education Area. This system is as follows:

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative mark may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Distinction (SB).

The distinction ‘Honours’ shall be awarded to students who have obtained a mark of 9.0 or higher. The number of students awarded this distinction may not exceed five per cent of those enrolled on the course in the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘First Class Honours’ may be awarded.

Timetable

Click on this link to view the detailed timetable in Excel

Bibliography

Basic:

1. A. Fernández Rañada

Classical Dynamics

Alianza. 1994.

ISBN: 84-206-8133-4

Supplementary:

2. J.R. Taylor

Classical Mechanics

Reverté. 2013.

ISBN: 8429143122

Others:

3. C. Kittel, W.D. Knight, M.A. Ruderman

Mechanics

Reverté. 1968.

ISBN: 978-84-291-42

4. S.T. Thornton, J.B. Marion

Classical Dynamics of Particles and Systems

Reverté. 1975.

ISBN: 9788429140941

Objectives

The aim of the module is to equip students with the necessary tools to understand thermodynamic processes and the principles and laws that govern them, so that they are able to successfully analyse and solve problems within the field of thermodynamics.

Prerequisites

None

Learning Outcomes

RK1 To be familiar with the most important phenomena and theories in the various branches of physics, as well as their historical context

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena described by them

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RS10 Apply the principles and laws of thermodynamics involved in the analysis of physical phenomena.

Learning outcomes

RA1 Identify the essential aspects of physical phenomena, describing them quantitatively and qualitatively using thermodynamic formalism

RA2 Presents and interprets thermodynamic information (graphs, tables, etc.)

RA3 Understands the First Law as a general principle of energy conservation, with a state function, namely internal energy.

LA4 States the Laws of Thermodynamics, analyses their implications and applies them to problem-solving

RA5 Understands how entropy and its properties account for the thermodynamic behaviour of systems.

RA6 Identifies thermodynamic potentials and analyses the thermodynamic behaviour of systems.

Description of the content

- The zeroth law. The concept of temperature.

- Thermodynamic relations.

- Fundamental thermodynamic equation.

- Thermodynamic processes

- First law: work, internal energy and heat. Enthalpy.

- Second law: entropy.

- Thermodynamic potentials, equilibrium and stability.

- Open systems, phase transitions, critical points.

- Third law.

Teaching activities

Training activity No. of hours* Contact hours

(8–12)** % Face-to-face

AP1.- Participatory lectures 24 4 100

AP2.- Seminars or practical application classes 15 2.5 100

AP3.- Practical activities (case studies, project work, simulations, etc.) 36 3 50

AP4. – Independent study 60 0 0

AP5.- Tutorials 12 0.6 30

AP6.- Knowledge assessments 3 0.5 100

Assessment system and criteria

Regular assessment period.

Assessment system | Weighting

SE1.- Practical activities | 25. Mid-term exam

SE2.- Final knowledge assessments | 60. Final exam (ordinary examination session)

SE4.- Portfolio | 15. This assessment system involves the daily submission of questions and problems not solved in class by the lecturer. This assessment system will be weighted against the student’s overall attendance for the module: if, at the end of the academic year, the student’s attendance at the scheduled class hours (SESSION, LAB) is 60 per cent or more, the mark for this assessment system will be that awarded following the lecturer’s marking. If the student’s attendance at all scheduled course sessions (SESSION and LAB) is less than 60 per cent, the mark for this assessment system, as determined following the lecturer’s marking, will be reduced in proportion to the level of attendance achieved, in accordance with an attendance criterion defined by the lecturer.

In order for the continuous assessment (comprising Practical Activities (SE1) and the Portfolio (SE4)) to be taken into account, students must

achieve a mark of 5.0/10 or higher in the final exam of the standard examination session. Otherwise, their mark will correspond directly to the

mark obtained in that exam.

The module is considered passed in the ordinary examination period if the final mark is 5.0/10 or higher.

Supplementary examination period

If a student has not passed the module in the ordinary examination period, they may sit the examination in the supplementary period. The supplementary examination period will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

100 per cent of the final mark for the module in the supplementary examination period is based on the final exam for this period.

The module is considered passed in the supplementary examination period if the final mark is 5.0/10 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

Objectives

The aim of this module is to provide students with a basic understanding of electrodynamics and its relationship with special relativity.

Prerequisites

No prerequisites

Learning Outcomes

RK1 To be familiar with the most important phenomena and theories in the various branches of physics, as well as their historical context

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena described by them

RK11 Understand the fundamentals and basic concepts of electric and magnetic fields, as well as the interrelationship between these fields and their unification in electromagnetism

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

Learning outcomes

RA5 Understands the relevant aspects of the propagation of electromagnetic waves in free space and in the presence of boundaries

RA6 Identifies the mechanisms of electromagnetic wave emission.

RA6 Analyses the phenomena of propagation and emission of electromagnetic waves.

LR7 Demonstrates an understanding of the close relationship between electromagnetism and the theory of relativity.

Course content

Topic 1: Conductors in electrostatic equilibrium

Topic 2: Electrodynamics

-Electromotive force

-Electromagnetic induction

-Maxwell’s equations

-Electromagnetic waves

Topic 3: Electromagnetism and relativity.

Topic 4: Electromagnetic radiation and radiating systems.

Teaching activities

AP1.- Interactive lectures

AP2.- Seminars or practical application classes

AP3. Practical activities (case studies, project work, simulation, etc.)

AP4. Independent study

AP5. Tutoring

AP6. Knowledge assessments

Assessment system and criteria

Continuous assessment

Activity Weighting (%)

SE1 – Practical activities 40

Mid-term exam (20%).

Submission of questions and problems not solved in class (20%).

SE2. Final knowledge assessments. 60

For component SE1 to count towards the final mark, a minimum mark of 4/10 must be achieved in the final exam during the standard examination period. In any other case, this component will be weighted at 0/10.

Extraordinary final assessment

In the supplementary sitting, the exam mark counts for 100 per cent of the final mark.

Bibliography

Core:

1. Griffiths, David J.

Introduction to Electrodynamics

Pearson. 2014.

ISBN: 9781292021423

2. Jackson,

J.D. Classical Electrodynamics,

Cambridge University Press. 2017.

ISBN: 9781119770763

Supplementary:

3.- R.P. Feynman, R.B. Leighton, M. Sands

The Feynman Lectures on Physics, Vol. II: The New Millennium Edition: Mainly Electromagnetism and Matter: 02

Inter-American Educational Fund. 1972.

ISBN: 9780465040841

Objectives

The aim of this course is to introduce students to experimentation in physics, data analysis and error propagation in the fields of mechanics, electromagnetism and thermodynamics.

Prerequisites

No prerequisites

Learning Outcomes

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of Physics.

RS3 Estimate orders of magnitude to interpret laboratory phenomena in the field of Physics and its sub-disciplines, as well as in Chemistry.

RS5 Use appropriate electronic instruments and/or computer tools in modelling to find solutions to physical problems.

Learning outcomes

RA1 Understands the principles, techniques and measuring instruments, as well as the phenomena of interest in Mechanics and Waves, Thermodynamics and Electromagnetism.

RA2 Uses measuring instruments appropriately and efficiently (Mechanics and Waves, Thermodynamics and Electromagnetism), following measurement protocols, particularly those relating to the safety of the experimenter.

LA3 Is able to assess the limitations of measurement methods due to interference, the simplicity of models and the effects that are neglected in the measurement method (Mechanics and Waves, Thermodynamics and Electromagnetism).

RA4 Plots data graphically, extracts information from the plot, analyses the data, models the results and compares them with physical laws relating to Mechanics and Waves, Thermodynamics and Electromagnetism.

RA5 Documents the measurement process in terms of its underlying principles, the instrumentation required and the conditions under which it is valid, carrying out a comprehensive analysis in accordance with the IMRD format (Mechanics and Waves, Thermodynamics and Electromagnetism).

RC1 Work independently on the management of projects related to the different areas of physics

Course description

Six experiments will be carried out:

Mechanics Labs:

Free fall

Thermodynamics Labs:

2. Ideal gas law

3. Thermal conductivity

4. Adiabatic gas law

Electromagnetism Labs:

5. Magnetic field in coils

6. Faraday’s law of induction

Training activities

AP1. – Participatory lectures

AP2. – Seminars or practical application classes

AP3. Practical activities (case studies, project work, simulations, etc.)

AP4. Independent study

AP5. Tutoring

AP6. Assessment tests

AP10.- Workshop and/or laboratory activities

Assessment system and criteria

1. Writing of laboratory reports (SE1 + SE3, 40%), one of which must be in article format (SE2, 20%):

a PDF produced using LaTeX with the provided template and written in English.

The remaining reports may be in any format preferred by the student, although LaTeX remains compulsory............. 60%

2. Laboratory notebook (SE3)............................................................................... 20%

3. Oral exam (SE2)............................................................ 20%

Objectives

The aim of this module is to provide students with a sound grounding in the fundamentals of mathematical logic, combinatorics, graph theory and number theory, equipping them to model and solve problems typical of discrete mathematics. By the end of the module, students will be able to:

1. Understand and apply the fundamentals of mathematical logic, including propositional and predicate logic, as well as rigorous methods of proof.

2. Develop structured mathematical reasoning to tackle abstract and applied problems in various contexts.

3. Analyse properties of integers using number theory, covering topics such as divisibility, modular arithmetic and congruences.

4. Understand and use relations and functions in the context of finite and infinite sets, applying them to the modelling of mathematical structures.

5. To study combinatorial structures and apply fundamental principles of combinatorics to the counting and enumeration of discrete configurations.

6. Formulate and solve problems in graph theory, including properties of graphs, paths, cycles and connectivity.

7. Apply knowledge of discrete mathematics to practical problems involving optimisation, algorithms and modelling in science and engineering.

8. Develop analytical and algorithmic skills for problem-solving using discrete approaches.

Prerequisites

No prerequisites

Competencies

RK6 Understand the principles of mathematics and statistics underpinning the study of physics in classical and quantum systems

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of Physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

RC1 Work independently on the management of projects related to the various areas of physics

RC3 Acquire IT knowledge and skills enabling the development of methods and technologies applicable to the relevant fields of study

RS5 Use appropriate electronic instruments and/or computer tools in modelling to find solutions to physics problems.

Learning outcomes

RA1 Appreciate the foundations of logic, mathematical proofs and algorithmic reasoning.

RA2 Formulates information using logical statements.

RA3 Applies the fundamentals of number theory to problem-solving.

RA4 Relates the basic properties of trees and graphs to examples from intelligent systems.

Course description

- Basic concepts. Propositional logic. Predicate logic. Inference in predicate logic.

- Sets. Relations and functions. Combinatorics.

- Lattices and Boolean algebras. Boolean functions.

- Number theory. Graph theory.

- Binary operations. Divisibility and modular arithmetic. Counts, basic structures and rules of combinatorics. Recurrence and induction. Lists and sets.

- Applications, equivalence and order relations. Combinatorial optimisation. Application examples using mathematical software (Python, Java or similar)

Teaching activities

AP1.- Participatory lectures

AP2. – Seminars or practical application sessions

AP3.- Practical activities (case study sessions, project work, simulation, etc.)

AP4.- Independent study

AP5. Tutorials

AP6. Knowledge assessments

Assessment system and criteria

ENG:

Continuous assessment during the regular term:

Group practical activities (10 per cent): Solving applied problems in the field of discrete mathematics and its applications.

Assignment submissions (10 per cent): Individual exercises to reinforce theoretical and practical concepts.

One mid-term exam (20 per cent): A minimum mark of 4 is required for it to be included in the average.

Regular session final exam (60%): A minimum mark of 4.0/10 in the regular session exam is required for it to be included in the final course mark.

In addition, continuous assessment activities will always count towards the final mark in the regular session.

Extraordinary session exam:

The exam mark will account for 100% of the course mark.

ESP:

Continuous assessment in the regular session:

Group practical activities (10 per cent): Solving applied problems in the field of discrete mathematics and its applications.

Submission of exercises (10 per cent): Individual exercises to consolidate theoretical and practical concepts.

A mid-term exam (20%): A minimum mark of 4 is required for the mark to count towards the final average.

Regular examination (60%). A minimum mark of 4.0/10 is required in the regular examination for it to be included in the final mark for the module.

Furthermore, continuous assessment activities will always be factored into the mark for the main examination.

Supplementary exam:

The exam mark will account for 100% of the course mark

Bibliography

Essential:

1. Johnsonbaugh, Richard

Discrete Mathematics

6th ed.: Pearson Education. 2005.

ISBN: 9702606373

Objectives

This course, together with ‘Mechanics and Waves I’, forms part of the Mechanics and Waves module. Its main objective is for students to become familiar with Lagrangian mechanics and Hamiltonian mechanics, two reformulations of Newtonian mechanics upon which much of modern fundamental physics is built (including Quantum Field Theory and General Relativity). In addition to understanding the conceptual foundations of these formalisms, students should be able to apply them to solving mechanical problems.

Furthermore, an introduction to the two-body problem will be presented, and the dynamics of particles interacting via central forces and the resulting orbits will be discussed. Finally, the implications of moving away from the classical notion of absolute space and time and adopting the postulates of Special Relativity in the development of a mechanical theory will be explored.

Prerequisites

No prerequisites

Learning Outcomes

RK1 To be familiar with the most important phenomena and theories in the various branches of physics, as well as their historical context

RK2 To understand physically distinct phenomena and their underlying analogies, enabling the application of known solutions to new problems

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RK5 Understand the scope and limitations of classical physics that led to the formulation of special and general relativity, as well as quantum mechanics, to address the new problems arising in modern physics.

RK7 Understand the laws and principles of physics, identifying their logical and mathematical structure, their experimental basis and the phenomena they describe

RK12 Understand the theories, laws and models governing physical phenomena related to mechanics

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

Learning outcomes

RA5 Applies the Lagrangian and Hamiltonian formulations appropriately to the solution of

mechanical problems.

RA6 Formulates the equations of motion for central forces, solving them completely and obtaining the solutions for the motion and the trajectories.

LA7 Formulates and solves the equations of a system that deviates from its equilibrium position, classifying said equilibrium.

RA8 Deepens their understanding of the fundamentals of special relativity and its most significant physical consequences, achieving proficiency in the study of particle kinematics and dynamics within the context of Minkowski spacetime.

Description of the content

- Lagrangian mechanics.

- The two-body problem. Central forces. Orbits.

- Hamiltonian mechanics.

- Relativistic mechanics.

Teaching activities

AP1.- Interactive lectures

AP2.- Seminars or practical application classes

AP3.- Practical activities (case studies, project work, simulation, etc.)

AP4. Independent study

Assessment system and criteria

ING

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competencies.

ASSESSMENT SYSTEMS

The assessment systems for this course are:

- SE1: Practical activities. These consist of two mid-term exams taken during the teaching period.

- SE2: Final knowledge assessment. This corresponds to a final exam covering the entire course syllabus.

- SE4: Portfolio. This assessment method involves the completion and submission of approximately four sets of exercises, which may be carried out individually or in small groups depending on the progress of the course.

ASSESSMENT CRITERIA

The assessment methods described above are set out in the following assessment criteria. There are two official examination sessions: ordinary and supplementary.

ORDINARY SESSION

The final mark for this session is the weighted average of the assessment components detailed below:

- Practical activities (SE1): 25 per cent of the final mark (12.5 per cent for each mid-term exam).

- Final knowledge assessment (SE2): 60 per cent of the final mark.

- Portfolio (SE4): 15 per cent of the final mark.

For the continuous assessment (comprising the portfolio and the two mid-term exams) to be taken into account, students must achieve a minimum mark of 4.0 in the final exam of the ordinary session. Otherwise, the final mark will correspond directly to the mark achieved in that exam.

The course is considered passed in the ordinary session if the final mark is 5.0 or higher.

EXTRAORDINARY SESSION

If a student does not pass the course in the ordinary session, they may sit the extraordinary session.

The supplementary session takes place during the July examination period (for further information, please refer to the Academic Calendar). It consists of a single examination covering the entire course content. The mark in the supplementary session corresponds directly to the mark obtained in this examination.

The course is considered passed in the extraordinary session if the final mark is 5.0 or higher.

GRADING SYSTEM

Article 5 of Royal Decree 1125/2003, of 5 September, establishes the grading system applicable to modules within degree programmes in the European Higher Education Area.

To obtain the corresponding credits, students must pass the exams or assessment tests associated with the course.

The level of learning achieved by students will be expressed using numerical marks on a scale from 0 to 10, to one decimal place, to which a qualitative mark may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: Excellent (SB).

The distinction ‘Honours’ (Matrícula de Honor) may be awarded to students who achieve a mark of 9.0 or higher. The number of such distinctions may not exceed five per cent of the students enrolled on the course during the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘Honours’ distinction may be awarded.

ESP

The assessment process will consist of verifying and evaluating the student’s acquisition of the required competences.

ASSESSMENT SYSTEMS

The assessment systems for this module are:

- AS1: Practical activities. These consist of two mid-term exams held during the term.

- AS2: Final knowledge assessments. These consist of a final examination covering the entire syllabus of the module.

- AS4: Portfolio. This assessment method involves completing and submitting approximately four sets of exercises, which may be undertaken individually or in small groups depending on the progress of the module.

ASSESSMENT CRITERIA

The assessment systems described above are set out in the following assessment criteria. There are two official examination sessions: the ordinary and the supplementary.

REGULAR SESSION

The final mark for this sitting is the weighted average of the assessment tests detailed below:

- Practical activities (SE1): 25% of the final mark (12.5% for each mid-term exam).

- Final knowledge assessments (SE2): 60% of the final mark.

- Portfolio (SE4): 15 per cent of the final mark.

For continuous assessment (comprising the portfolio and the two mid-term exams) to be taken into account, students must achieve a minimum mark of 4.0 in the final exam of the ordinary assessment period. Otherwise, their mark will correspond directly to that obtained in that exam.

The module is considered passed in the ordinary examination period if the final mark is 5.0 or higher.

SUPPLEMENTARY EXAMINATION SESSION

If a student has not passed the module during the regular examination period, they may sit the supplementary examination.

The supplementary examination period will take place during the July examination period (for further information, please refer to the Academic Calendar). It consists of a single examination covering the entire syllabus of the module. The mark for the supplementary examination corresponds directly to the mark obtained in this examination.

The module is considered passed in the extraordinary examination session if the final mark is 5.0 or above.

GRADES

Article 5 of Royal Decree 1125/2003 of 5 September establishes the marking system applicable to modules within degree programmes falling within the scope of the European Higher Education Area. This system is as follows:

The award of the corresponding credits is conditional upon passing the associated examinations or assessment tests.

The level of learning achieved by students will be expressed as numerical marks on a scale of 0 to 10, to one decimal place, to which the corresponding qualitative grade may be added:

- 0–4.9: Fail (SS).

- 5.0–6.9: Pass (AP).

- 7.0–8.9: Good (NT).

- 9.0–10: High distinction (SB).

The distinction ‘Honours’ shall be awarded to students who have obtained a mark of 9.0 or higher. The number of students awarded this distinction may not exceed five per cent of those enrolled on the course in the relevant academic year, unless the number of enrolled students is fewer than 20, in which case only one ‘First Class Honours’ may be awarded.

Bibliography

Essential:

1. A. Fernández Rañada

Classical Dynamics

1st ed. Alianza. 1994.

ISBN: 8420681334

2. H. Goldstein

Classical Mechanics

Addison-Wesley. 1950.

ISBN: 9780201025101

3. J.R. Taylor

Classical Mechanics

Reverté. 2013.

ISBN: 8429143122

4. S.T. Thornton, J.B. Marion

Classical Dynamics of Particles and Systems

Reverté. 1975.

ISBN: 9788429140941

Objectives

To understand, study and apply theoretical and practical results relating to analytic functions, in particular elementary functions and their compositions. Understand and apply the Cauchy–Gousart theorem and the Cauchy integral formula, in its various forms for functions of a complex variable, to the integration of holomorphic functions. Understand and correctly apply Cauchy’s residue theorem and its applications. Understand the discrete Fourier transform and its properties, and apply it to signal theory: the fast Fourier transform and signal filtering; and apply it to image processing and audio compression.

Prerequisites

No prerequisites.

Learning Outcomes

RK2 Understand physically distinct phenomena and their underlying analogies in order to apply known solutions to new problems

RK6 Understand the principles of mathematics and statistics that underpin the study of physics in classical and quantum systems

RS1 Apply the most important knowledge, concepts and methods from the various branches of physics.

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting the appropriate tools and interpreting results.

Learning outcomes

LA1 Solve problems involving complex functions of a complex variable and integration in the complex plane.

RA2 Explains the concepts of series convergence, singularities and residues.

RA3 Applies the residue theorem in calculating integrals to solve physics problems.

LA4 Carries out work by applying knowledge of and using the elementary complex functions that appear in various areas of physics.

LA5 Applies the Fourier transform to solve physics problems.

Course content

- Complex numbers. The complex plane. Complex functions of a complex variable. Analytic functions and their properties.

- Elementary complex functions and transformations.

- Integration in the complex plane. Cauchy’s theorem. Series, singularities and residues.

- Applications of the residue theorem.

- Fourier series and their applications

- Fourier analysis. Integral Fourier transform and applications.

Teaching activities

AP1.- Interactive lectures

AP2. Seminars or practical application sessions

AP3. Practical activities (case study sessions, project work, simulation, etc.)

AP4. – Independent study

AP5. – Tutorials

AP6. Knowledge assessments

Assessment system and criteria

Compulsory continuous assessment consists of two written examinations. Calculators and formula sheets are not permitted. These assessments do not exempt students from further coursework. They each account for 15 per cent of the mark. Students will also be required to submit practical exercises on a specific section of the course, which account for 10 per cent of the mark. Students will also sit the official exam in the ordinary assessment period, which has the same format as the tests and accounts for 60 per cent of the mark.

Students sitting the supplementary examination will take a single exam, which accounts for 100% of the mark.

Objectives

This module will lay the foundations of electronics and electronic circuits, providing students with the knowledge required to analyse electronic circuits and understand their uses and applications. Furthermore, it will introduce students to electronic design and its applications in everyday life.

The most important electronic components will be studied, beginning with RLC circuits, covering both direct current and alternating current, followed by operational amplifiers and diodes, and concluding with BJK transistors and MOSFETs. Particular emphasis will be placed on the applications of these circuits, such as the description of power supply circuits and the main logic gates.

Prerequisites

None

Learning Outcomes

RK1 Understand the most important phenomena and theories in the various branches of physics, as well as their historical context

RK3 To analyse the fundamental concepts and principles of physical systems in order to develop approximations that enable the construction of a simplified model

RC4 Understand the processes involved in obtaining various types of materials, their physical fundamentals and their applications.

Learning outcomes

RA1 Analyses direct current and alternating current circuits.

RA2 Understands the fundamental devices – diodes and bipolar and field-effect transistors – and their description using simple functional models.

RA3 Understands the main applications of the transistor in amplifier circuits: basic amplifier circuits and operational amplifiers.

LA4 Designs circuits using operational amplifiers.

LA5 Understands and uses electronic simulation software.

RA6 Understands the applications of the transistor in digital electronics.

Course content

- Ohm’s Law and Kirchhoff’s Laws; DC circuit analysis.

- Analysis of RLC circuits. AC circuits. Transient phenomena; analysis of AC circuits. Steady-state phenomena.

- Network analysis: Thevenin and Norton methods

- Biasing. Small-signal equivalent model. Single-stage amplifiers. Frequency response. Cascading of amplifier stages.

- Operational amplifiers and applications. Ideal operational amplifier.

- Amplifiers. Equivalent circuits of amplifiers. Feedback. Frequency-domain analysis. Amplification stages.

- Introduction to semiconductor theory. p-n junction. Diode, bipolar junction transistor (BJT) and MOSFET. Characteristic equations.

Amplifying function of the BJT and the MOSFET.

Learning activities

Learning activity No. of hours* Contact hours

(8–12)** % Face-to-face

AP1.- Participatory lectures 24 4 100

AP2.- Seminars or practical application classes 15 2.5 100

AP3.- Practical activities (case studies, project work, simulations, etc.) 36 3 50

AP4. – Independent study 60 0 0

AP5.- Tutorials 12 0.6 30

AP6.- Knowledge assessments 3 0.5 100

TOTAL 150 10.6

Assessment system and criteria

Regular examination session

The final mark for this examination session is the weighted average of a set of assessment tests detailed below:

Continuous assessment (40 per cent):

SE3 – Portfolios: 10%

SE1 – Case studies/problem-solving: 30%

Final assessment (60%):

SE2 – Final knowledge tests: 60% (A minimum mark of 4/10 is required for the continuous assessment to be taken into account. Otherwise, the mark will correspond directly to that obtained in the exam.)

The module is considered passed in the standard assessment period if the final mark, taking all of the above into account, is 5.0/10 or higher.

Supplementary examination

If a student has not passed the module in the ordinary examination session, they may sit the supplementary examination session. The supplementary examination session will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

100 per cent of the final mark for the module in the supplementary sitting corresponds to the final examination for this sitting, which assesses the entire syllabus of the module.

The module is considered passed in the resit period if the final mark is 5.0/10 or higher.

Timetable

Click on this link to view the detailed timetable in Excel

Bibliography

Basic:

1. Charles, K., & Alexander, S.

Fundamentals of Electrical Circuits

McGraw-Hill Interamerican, 2013.

ISBN: 978-607-15-09

Objectives

- Identify and implement standard abstract data types (ADT): lists, stacks, queues, trees, graphs, hash tables.

- Analyse algorithm efficiency using asymptotic notation (Big-O, Ω, Θ).

- Apply fundamental algorithm design paradigms: greedy, divide-and-conquer, and dynamic programming.

- Read, write and process scientific file formats (CSV, Excel, GeoTIFF, shapefile, NetCDF, GRIB).

- Design and query relational databases (SQLite) and understand NoSQL models.

- Solve problems in the physical and environmental sciences using the appropriate data structure and algorithm.

- Identify and implement standard abstract data types (ADTs): lists, stacks, queues, trees, graphs, hash tables.

- Analyse the efficiency of algorithms using asymptotic notation (Big-O, Ω, Θ).

- Apply fundamental algorithm design paradigms: greedy, divide and conquer, and dynamic programming.

- Read, write and process scientific file formats (CSV, Excel, GeoTIFF, shapefile, NetCDF, GRIB).

-Design and query relational databases (SQLite) and understand NoSQL models.

-Solve problems in the physical and environmental sciences using the appropriate data structures and algorithms.

Prerequisites

None

Competencies

RK6 Understand the principles of mathematics and statistics that underpin the study of physics in classical and quantum systems. / Know the principles of mathematics and statistics that support the study of physics in classical and quantum systems.

RS2 Carry out calculations, assessments, studies, reports and tasks to produce high-quality work in the field of physics. / Conduct calculations, assessments, studies, reports and tasks to develop high-quality work in the field of physics.

RS4 Apply mathematical and numerical methods to the modelling and explicit solution of problems in physics and related disciplines, selecting appropriate tools and interpreting results. / Apply mathematical and numerical methods in the modelling and explicit resolution of problems in physics and related disciplines, selecting appropriate tools and interpreting results.

RS6 Prepare reports, written works and other scientific documents in the field of physics, communicating them clearly and effectively both in writing and orally, also using the English language. / Prepare reports, written works and other scientific documents in the field of physics, communicating them clearly and effectively both in writing and orally, also using the English language.

RC1 Carry out independent work in the management of projects related to the various areas of physics. / Develop independent work in the management of projects related to the various areas of physics.

RC3 Acquire computer knowledge and skills that enable the development of methods and technologies applicable in the relevant fields of study. / Acquire computer knowledge and skills that enable the development of methods and technologies applicable in the relevant fields of study.

Learning outcomes

RA1 Understands the various models and data structures and develops algorithms appropriate to the data, depending on the requirements of the problem to be solved

RA2 Applies knowledge of algorithms and computational complexity to solve problems in physical systems and related fields

RA3 Identifies and proposes solutions to problems relating to algorithm efficiency

LA4 Designs and scales algorithms for environments of varying size and complexity

RA5 Solves problems that may arise in physics by applying knowledge

relating to the structure and programming of computer systems

Course content

- Introduction, architecture, design and database models.

- Linear and associative structures (ADT).

- Hierarchical and network structures (trees, heaps, graphs).

- Data structures on disk.

- Analysis of algorithm efficiency.

- Algorithm design: greedy, divide-and-conquer.

- Introduction to dynamic programming.

- Introduction, architecture, design and models of databases.

- Linear and associative structures (TDA).

- Hierarchical and network structures (trees, heaps, graphs).

- Data structures on disk.

- Analysis of algorithm efficiency.

- Algorithm design: greedy, divide and conquer.

- Introduction to dynamic programming.

Learning activities

Learning activity No. of hours* Contact hours

(8–12)** % Face-to-face

AP1.- Participatory lectures 24 4 100

AP2.- Seminars or practical application classes 15 2.5 100

AP3.- Practical activities (case studies, project work, simulations, etc.) 36 3 50

AP4. – Independent study 60 0 0

AP5.- Tutorials 12 0.6 30

AP6.- Knowledge assessments 3 0.5 100

TOTAL 150 10.6

Assessment system and criteria

Assessment

Regular Examination Period

SE1: | Mid-term exam | 10% | | Coursework and final presentation | 20% |

SE2: | Final exam | 60% |

SE4: | Portfolio | 10% |

For continuous assessment (comprising SE1 and Portfolio SE4) to be taken into account, students must achieve a mark of 4.0/10 or higher in the final examination during the regular examination period. Otherwise, their mark will correspond directly to the mark achieved in that examination.

*The coursework and exposition (P1–P4) consists of take-home practical work (P1–P4), which is assessed alongside a course project defence at the end of the semester.

*The portfolio is a compilation of the formative work completed throughout the term.

The course is deemed to have been passed during the regular examination period if the final mark is 5.0/10 or higher.

Extraordinary Examination Period

If a student has not passed the course during the regular examination period, they may sit the supplementary examination. The supplementary examination will take place during the June and July examination period (for further information, please consult the Academic Calendar).

100 per cent of the final course mark during the supplementary examination period is based on the final examination for this period.

The course is considered passed during the extraordinary examination period if the final mark is 5.0/10 or higher.

Regular Examination Period

SE1: Mid-term exam | 10 %

SE1: Coursework and final presentation | 20 %

SE2: Final examination | 60 %

SE4: Portfolio | 10 %

For continuous assessment (comprising SE1 and the SE4 portfolio) to be taken into account, students must achieve a mark of 4.0/10 or higher in the final exam during the ordinary assessment period. Otherwise, the final mark will correspond directly to the mark obtained in that exam.

The course assignments and presentation (P1–P4) consist of practical tasks to be completed at home (P1–P4), which are assessed alongside the defence of a course project at the end of the semester.

The Portfolio is a compilation of the coursework completed throughout the academic term.

The module is considered to have been passed during the ordinary assessment period if the final mark is 5.0/10 or above.

Supplementary Assessment Period

If a student has not passed the module during the ordinary assessment period, they may sit the supplementary examination. The supplementary examination will take place during the June and July examination periods (for further information, please consult the Academic Calendar).

100 per cent of the final mark for the course during the supplementary assessment period is based on the final exam taken during this period.

The module is considered to have been passed during the supplementary assessment period if the final mark is 5.0/10 or higher.

Timetable

Click on this link to view the detailed timetable in Excel