Mathematical Methods I

Within the context of the curriculum of the Degree in Physics, Mathematical Methods I belongs to module 4 (Mathematical Methods for Physics), which has been allocated with 40.5 ECTS. The study and command of mathematics is of fundamental importance in science, specially for a physicist. Almost all branches of mathematics are relevant to physics, but if we had to highlight one due to its high horizontal impact in all areas of physics, that would be differential calculus.

Mathematical Methods I is an introductory course to differential calculus. Its aim is, first, to strengthen the knowledge on the foundations of calculus that the students bring from the high school and, second, to expand it in order to give them access to the concepts and mathematical tools needed in advanced calculus.

Regarding the goals of the learning process, the student should:
o Know the properties of real and complex numbers.
o Master the basic elements of point set topology.
o Be able to evaluate limits and continuity of functions.
o Compute derivatives of functions.
o Understand, both from a rigorous and an intuitive point of view, the limit of sequences and the sum of series.

• Chapter 1: The real and complex numbers systems.
o Axiomatic definition of the real numbers.
o The sets of integers, rational and irrational numbers.
o Finite decimal approximations to real numbers.
o Construction and properties of complex numbers.

•Chapter 2: Elements of point set topology. Topology of the real line and multidimensional spaces.
o Normed, metric and topological spaces.
o Open and closed sets.
o Interior points, points of accumulation and adherent points.
o compact sets.

• Chapter 3: Limits and continuity.
o Convergent sequences and Cauchy sequences.
o Complete metric spaces.
o Limits of functions.
o Continuous functions and homeomorphisms
• Chapter 4: Differential calculus.
o Derivatives of functions of real variable.
o Rolle's theorem. Mean Value Theorem.
o Taylor's formula for real functions of a real variable.

• Chapter 5: Series.
o Infinite series. Alternating series.
o Conditional convergence and absolute convergence.
o Convergence criteria.
o Functional series. Power series.

Basic bibliography:
• T.M. Apostol, Análisis Matemático, 2ª ed., Ed. Reverté, Barcelona.
• J.A. Fernández Viña, Análisis Matemático, Vol. 1,2,3, 2ª ed., Ed. Tecnos, Madrid.
• R.E Larson, R.P. Hostetler, Cálculo y Geometría Analítica. Ed. McGraw-Hill.
• J. de Burgos, Cálculo Infinitesimal de una Variable. Ed. McGraw-Hill.
• K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical methods for physics and engineering: a comprehensive guide. Cambridge University.

Complementary bibliography:
• T.M. Apostol, Calculus, 2ª ed., Vol. 1,2. Ed. Reverté.
• M. Spivak, Calculus, 2ª ed., Ed. Reverté.
• M. de Guzmán y B. Rubio, Problemas, conceptos y métodos del análisis matemático, Vols. 1, 2 y 3. Ed. Pirámide.
• W. Rudin, Principios de Análisis Matemático. Ed. McGraw-Hill.

Exercise Books:
• B. Demidovich, Problemas y Ejercicios de Análisis Matemático. Ed. Mir, Moscú.
• F. Bombal, L. Rodríguez, G. Vera, Problemas de Análisis Matemático, Vols 1,2,3. Ed. AC, Madrid.
• J.A. Fernández Viña, E. Sánchez Mañes, Ejercicios y Complementos de Análisis Matemático, Vols. 1,2, 4ª ed. Ed. Tecnos, Madrid.
• M. Spivak, Suplemento del Calculus. Ed. Reverté.

• BASIC AND GENERAL
The students will:
o CB1 – Have absorbed and understood essential material from a field which is one of the founding stones of high-school and college education. This material is usually found in advanced textbooks, but it includes also some elements coming from forefront research.
o CB2 – Be able to use their newly acquired knowledge to other fields in a professional way. They will show an increase of those skills that come naturally from heavy practice in complex reasoning and problem solving.
o CB5 – Have developed the learning skills needed to undertake further studies with a high degree of autonomy.
o CG3 – Be able to apply the newly acquired theoretical and practical knowledge to solve standard problems and pose original ones. Be able to use their strengthened analysis and abstraction skills as a tool to solve new problems in academic and professional contexts.
• TRANSVERSAL
The students will:
o CT1 – Have increased their analysis and synthesis abilities.
o CT2 - Have developed their skills for planning and organization.
o CT5 - Have developed critical thinking.
• SPECIFIC
The students will:
o CE5 – Be able to evaluate the complexity of a problem. When intractable as originally posed, try to find allowable approximations to make it affordable. They will show enough critical thinking to build simple a physical model to simulate the complexity of the real world.
o CE6 - Understand and master the mathematical and numerical methods most commonly used in Physics
o CE8 - Be able to efficiently handle bibliography as well as any source of relevant information, and use it both in research and technical projects.
In addition, the subject teachers will provide notes and bulletins of solved problems for each of the topics on the Virtual Campus. This material may be complemented with exam exercises from previous courses, electronic bibliographic material that may be acquired through the USC library after evaluation and request to the subject's professors, or online resources that meet the required quality standards. whose links will be provided to students through the Virtual Campus.

A course will be activated on the Moodle platform of the Virtual Campus, to which information of interest to the student as well as various teaching material will be uploaded.

The general methodological instructions given in the Report of the Degree in Physics of the USC will be followed. In particular, the teaching will be distributed per week in two master lectures plus another two lectures of interactive type where the students will be divided in small groups. Besides an individualized tutoring agreed with the students will be also given. The basic theoretical aspects of the course will be explained in the master lectures, while the interactive ones will be dedicated to solve problems and exercises with an active participation by the students (although eventually they could also be used to expand the contents of the master lectures). In the tutorials, it will be offered personalized attention to the needs of each student.

Tutorials may be in-person or telematic, if they are telematic they will require an appointment, which is also recommended for in-person tutorials.

The student evaluation will consist of two parts.
• The progress of each student in the assimilation of the subject will be continuously monitored through short written tests and/or the resolution of exercises individually or in groups in class. Discussion with the student of the conceptual aspects of the subject contained in said exercises and tests will be encouraged. The grade for this part will mean up to 2.5 points that will be added directly to the final exam grade, as long as the student has attended a minimum of 80% of the face-to-face classes (expository and interactive).
• There will be a final exam of eminently practical content that will consist of short questions and/or problems and will be held on the official dates set by the center.
• By adding the continuous evaluation grade directly to the result of the final exam, the stipulations of the Grade Report are fulfilled, which establishes that the final grade of the subject cannot be lower than that of the final exam. In cases of fraudulent completion of exercises or tests, the provisions of the Regulations for evaluating the academic performance of students and reviewing qualifications will apply.

The working time in presence of the teacher is 60 hours, classified as follows:
• 32 hours of master lectures in a large group.
• 24 hours of interactive lectures in small groups.
• 4 hours of tutoring for every student.
The autonomous and individual work time that the student should carry out to achieve the required knowledge on the subject will not exceed 90 hours.

It is of particular importance to acquire a regular habit of study of the subject and to devote a certain percentage of the hours of each working day to this course. It is also necessary for the student to be able to solve all the proposed problems and exercises by himself/herself. To read the solutions proposed by another person is not enough