ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Particle Physics
Areas: Theoretical Physics
Center Faculty of Physics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable
The general objective of the course is to provide the student with the basic mathematical tools for the study of physical problems. More specifically, the course has a fundamental block dedicated to the differential and integral calculus of functions of complex variables. Integral transforms of Fourier and Laplace are also studied. In the last part of the course, generalized functions will be studied.
This course is the last one in mathematical methods addressed in the degree. The focus of the course will be eminently practical and oriented towards calculation. Therefore, formal developments will be avoided as much as possible, and the practical applications of the different theoretical developments will be addressed as quickly as possible, ensuring that the problems studied are relevant in different fields of Physics.
Learning outcomes:
With respect to the subject Mathematical Methods VI, the student will demonstrate:
- Mastery, at a practical level, of the mathematical tools and calculation techniques necessary for the analysis and solution of physical problems.
- Furthermore, they will have acquired sufficient maturity to be able to address mathematical problems needed in their Physics studies effectively.
The course will be developed in accordance with the following syllabus:
THE COMPLEX PLANE. The field of complex numbers. Polar form and
complex exponentials. Roots of complex numbers. Topology of
the complex plane.
FUNCTIONS OF COMPLEX VARIABLE. Single-valued and multivalued functions:
Branches and Riemann surfaces. Analytic functions and Cauchy-
Riemann equations. Poles and branch cuts.
THE COMPLEX INTEGRAL. Cauchy's Theorem. Application to the calculation of real valued
integrals. Sum of series.
CAUCHY INTEGRAL FORMULAS. Theorems of Morera and Liouville. Fundamental
theorem of algebra. Argument Theorem. Laurent series.
INTEGRAL TRANSFORMS. Fourier Transform and its inverse.
Convolutions. Laplace transform. Application to solving differential and integral
equations.
GENERALIZED FUNCTIONS. The Dirac delta function and its derivatives.
Generalized Fourier transforms.
Basic Bibliography (includes the signature of the books in the library of the Department of Physics).
- M.R. Spiegel, Variable compleja, Ed. McGraw Hill (3 A02 59).
-R. V. Churchill, J. W. Brown, Variable compleja y aplicaciones, Ed. McGraw Hill (3 A02 163).
-R. Seely, Introducción a las series e integrales de Fourier, Ed. Reverte (3 A02 164).
-M.J. Lighthill, Introduction to Fourier análisis and generalizad functions, Ed. Cambrigde University Press.
Online Resources
There are many complex variables courses online. Some of them are:
http://math.fullerton.edu/mathews/complex.html
(Complex Analysis Project for Undergraduate Students, California State Univ., USA)
http://web.me.com/paulscott.info/CA2/contents.html
(Complex Analysis notes and interactive quizzes, University of Adelaide, Australia).
http://faculty.gvsu.edu/fishbacp/complex/complex.htm
(Grand Valley State University, Allendale, Michigan, USA)
The videos of the classes at Stanford University and MIT on Fourier transforms can be viewed at the following addresses:
http://www.cosmolearning.com/courses/the-fourier-transforms-and-its-app…
http://www.cosmolearning.com/video-lectures/filters-fourier-integral-tr…
http://www.cosmolearning.com/video-lectures/fourier-integral-transform-…
Informatio on Laplace trasforms can be found in
http://sites.science.oregonstate.edu/math/home/programs/undergrad/Calcu…
Information on sums of series can be found in
http://www.supermath.info/InfiniteSeriesandtheResidueTheorem.pdf
Information on generalized functions can be found in
https://cds.cern.ch/record/1453294/files/978-3-642-23617-4_BookBackMatt…
A useful reference is the website of Wolfram MathWorld:
http://mathworld.wolfram.com
Basic and general:
- That the students have showed to possess knowledge in an area of study that starts from secondary education, and is used to reach a level that, although supported in advanced textbooks, also includes the most recent findings in the field.
- That the students know to apply their knowledge to their professional work or their vocation and possess the relevant competences through the preparation and defence of arguments and the resolution of problems inside their area of study.
- That the students have developed those skills require to continue their studies with a high degree of autonomy.
- That they are able to apply both the theoretical and the practical knowledge like the capacity of analysis and of abstraction in the definition and approach to problems, and in the search of solutions both in and outside academia.
Transversal:
- Acquire capacity of analysis and synthesis.
- Acquire capacity of organisation and planning.
- Develop critical reasoning.
Specific:
- Be able to capture the essentials of a process or situation and develop a model, as well as perform the approximations required to reduce the problem to a handy level. They will show to possess critical thought to build physical models.
- Comprise and dominate the use of the mathematical and numerical methods more commonly used in Physics.
- Be able to search and use bibliography, as well as any source of relevant information, and apply it to research and technical development of projects.
A course in the Moodle platform in the Virtual Campus will be activated, where useful information and teaching material will be uploaded.
There will be lectures and classes of exercises and problems, both being face to face. Individual sessions with the professor will be either remote or in person, if remote they will require booking which is also advised for the ones in person.
During the course the student will be evaluated by performing a small number of controls, tests and proposed exercises. A qualification NC will come from this evaluation. At the end of the course, there will be a final exam consisting of solving problems or exercises, leading to qualification NE. The final qualification will be obtained using the formula MAX(0.4*NC+0.6*NE,NE) if NE is greater than or equal to 3.0. If NE is smaller than 3.0, NE will be used as final qualification.
Unethical conduct during any exercise or exam required for the assessment of the subject will imply the failure to pass that subject, irrespective of other disciplinary measurements that can be prosecuted against the infractor. Unethical behaviour will include, among others, the elaboration of memories copied or obtained from publicly available sources without re-elaboration or reinterpretation, and without citation of authors and sources.
The on-site classes will be 32 of theory, 24 of practice and 4 of tutorship. It is rather difficult to estimate the study time needed to assimilate the contents of the course, since it strongly depends on the dedication and capabilities of the student. As a general indication, in the report of the grade in physics of the USC the time for the personal work of the student is estimated to 75 hours. To this time we should add 15 hours needed to carry out the work in the lectures and other practical exercises. This will give a total of 90 hours.
The student needs a good knowledge of mathematical analysis of real variable and ease of use of elementary algebraic methods.
These skills are taught in the Mathematical Methods courses prior to this one.
Nestor Armesto Perez
Coordinador/a- Department
- Particle Physics
- Area
- Theoretical Physics
- Phone
- 881814107
- nestor.armesto [at] usc.es
- Category
- Professor: University Professor
Jose Daniel Edelstein Glaubach
- Department
- Particle Physics
- Area
- Theoretical Physics
- Phone
- 881813975
- jose.edelstein [at] usc.es
- Category
- Professor: University Lecturer
Xoan Mayo Lopez
- Department
- Particle Physics
- Area
- Theoretical Physics
- xoan.mayo.lopez [at] usc.es
- Category
- Ministry Pre-doctoral Contract
Sergio Barrera Cabodevila
- Department
- Particle Physics
- Area
- Theoretical Physics
- sergio.barrera.cabodevila [at] usc.es
- Category
- Xunta Pre-doctoral Contract
Francisco Sanchez Rodriguez
- Department
- Particle Physics
- Area
- Theoretical Physics
- franciscosanchez.rodriguez [at] usc.es
- Category
- Ministry Pre-doctoral Contract
| Tuesday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_01 | Galician, Spanish | Classroom 0 |
| 18:00-19:00 | Grupo /CLE_02 | Galician, Spanish | Classroom 6 |
| Wednesday | |||
| 11:00-12:00 | Grupo /CLE_01 | Galician, Spanish | Classroom 0 |
| 18:00-19:00 | Grupo /CLE_02 | Galician, Spanish | Classroom 6 |
| Thursday | |||
| 11:00-12:00 | Grupo /CLE_01 | Galician, Spanish | Classroom 0 |
| 18:00-19:00 | Grupo /CLE_02 | Spanish, Galician | Classroom 6 |
| Friday | |||
| 11:00-12:00 | Grupo /CLE_01 | Spanish, Galician | Classroom 0 |
| 18:00-19:00 | Grupo /CLE_02 | Galician, Spanish | Classroom 6 |
| 05.14.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 0 |
| 05.14.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 130 |
| 05.14.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 6 |
| 05.14.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 830 |
| 06.27.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 0 |
| 06.27.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 6 |
| 06.27.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 830 |