Mathematical Methods V

This subject is part of the module "Mathematical Methods of Physics", with the general objective of providing students with the mathematical tools necessary for the development and understanding of the different branches of Physics. The specific objectives of the subject are:

• Apply differential and integral calculus to the study of curves and surfaces in three-dimensional Euclidean space.
• Introduce the basic concepts and methods necessary for the study and resolution of partial differential equations of physical interest.

Learning outcomes:

After completing the subject, the student:
• Will know how to write curves and surfaces using the concepts and methods of differential calculus.
• Will understand the basics of eigenvalues and eigenfunctions problems.
• Will be familiar with the main methods to solve equations in partial derivatives of physical interest.

Curves and Surfaces:
• Parameterized curves and regular curves. Arc-length. Curvature and torsion. Frenet-Serret formulas: intrinsic equations.
• Differentiable surfaces. Tangent plane and normal line. First and second fundamental forms: main curvatures.

Second Order differential equations: Boundary value problems:
• Sturm-Liouville problems: eigenvalues, eigenfunctions and generalized Fourier series.
• Non-homogeneous problems: Green's function.

Equations in Partial Derivatives:
• First order partial differential equations: Characteristic curves.
• Second order partial differential equations: Classification and canonical forms. Initial and boundary conditions. Wave equation. Laplace equation. Poisson equation. Heat equation. Method of separation of variables.

Basic bibliography:

• Shifrin, T .: Differential Geometry: a First Course on Curves and Surfaces, Univ. Of Georgia 2016. http://alpha.math.uga.edu/ shifrin / ShifrinDiffGeo.pdf
• Carmo, M. P. do: Differential geometry of curves and surfaces. Editorial Alliance, Madrid 1995.
• Lipschutz, M.M.: Differential geometry. McGraw-Hill, Madrid 1990.
• Simmnons, G. F.: Differential equations, McGraw-Hill, Madrid 1988.
• Riley, K. F., Hobson, M. P. and Bence, S. J.: Mathematical methods for physics and engineering, Cambridge University Press, 2006.
• Duchateau, P. and Zachmann, D. W.: Partial differential equations. McGraw-Hill, Mexico 1988.
• Haberman, R.: Equations in Partial Derivatives, 3rd edition. Prentice Hall, Madrid 2003.
• Stephenson, G: Partial Differential Equations outside Scientists and Engineers, Longman, New York, 1985.
• Aranda, P: Notes on Mathematical Methods II (EDPs), UCM, 2017. http://jacobi.fis.ucm.es/pparanda/edps.html
• Myint- U, T. and Debnath, L: Linear partial differential equations outside scientists and engineers, Birkhauser, Boston, 2007.


Further reading:

• Hartle, J. B.: Gravity: and introduction to Einstein’s General Relativity, 2003.
• Arfken, G.: Mathematical Methods for Physicists, 3rd edition. Academic Press, 1985.
• Stone, M. and Goldbart, P.: Mathematics for Physics. Cambridge University Press, 2009.

Online resources:

• Virtual Classroom: Notes made by the teachers: Differentiable curves, Differentiable surfaces, Second order linear differential equations, Differential equations in partial derivatives.
• Virtual Classroom: Problem bulletins, problem solutions, exams from previous courses.
• Virtual Classroom: Links to online resources.

BASIC AND GENERAL

CB1 - Demonstrate to possess and understand knowledge of an area of study that begins at the base of general secondary education, and is usually found at a level that, although supported by advanced textbooks, also includes some aspects that involve knowledge from the forefront of their field of study.
CB2 - Students will know how to apply their knowledge to their work or vocation in a professional way and possess the competencies that are usually demonstrated through the elaboration and defense of arguments and the resolution of problems within their area of study.
CB3 - Have the ability to collect and interpret relevant data (usually within their area of study) to make judgments that include reflection on relevant issues of a social, scientific or ethical nature.

CG3 - Apply both the theoretical-practical knowledge acquired and the capacity for analysis and abstraction in the definition and approach of problems and in the search for solutions in both academic and professional contexts.

TRANSVERSAL

CT1 - Acquire analysis and synthesis capacity.
CT2 - Have organization and planning capacity.
CT5 - Develop critical reasoning.

SPECIFIC

CE2 - Be able to clearly manage orders of magnitude and make adequate estimates in order to develop a clear perception of situations that, although physically different, show some analogy, allowing the use of known solutions to new problems.
CE5 - Be able to carry out the essential of a process or situation and establish a working model of it as well as perform the required approximations in order to reduce the problem to a manageable level. Demonstrate critical thinking to build physical models.
CE6 - Understand and master the use of the most commonly used mathematical and numerical methods in Physics.
CE8 - Be able to manage, search and use bibliography, as well as any relevant source of information and apply it to research and technical development of projects.

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

The general methodological indications established in the USC Degree Physics Report will be followed. Teaching will be face-to-face and is scheduled in theoretical classes (32 hours), small group practices (24 hours) and tutoring in very small groups (4 hours).
In the theoretical and practical classes the basic contents of the subject will be presented and some exercises will be solved.
The most advanced contents and problems will be proposed to the student for their personal work with the support of the tutoring hours.
The tutorials may be face-to-face or telematic, if they are telematic they will require an appointment, which will also be recommended for face-to-face.

The qualification of the students consists of two parts:
• A final face-to-face exam to be held on the official dates set by the center, with a NEF score.
• The continuous evaluation that will consist of:
- Attendance and active participation in face-to-face classes and tutorials, as well as delivery of exercises. This part will lead to the APC grade.
- Carrying out two face-to-face written controls that allow evaluating the skills and knowledge acquired. These controls will result in a C rating

The joint score of these continuous assessment activities will lead to a “course grade” NC = APC*0.35 + C*0.65.

The final grade (NOTE) will be obtained as the maximum between the final exam grade and the result of averaging the NC course grade with the grade obtained in the NEF final exam using the formula NOTE = NEF*0.6 + Max (NC, NEF )*0.4.

This system will be applied both in the first and in the second opportunity.

Continuous assessment will only be valid during the academic year, and will not be kept for subsequent years.

In cases of fraudulent performance of exercises or tests, the one set forth in the Regulations for the evaluation of the academic performance of students and the review of grades will be applied .

n.

32 hours of face-to-face or telematic lectures.
24 hours of interactive classroom or telematic classes.
4 hours of face-to-face or telematic tutoring.

It is difficult to determine the study time necessary to assimilate the subject, since it depends a lot on the dedication and ability of each student. As a general indication, in the USC's Memory of the Degree in Physics, the student's personal work is estimated at 75 hours, not counting face-to-face or telematic teaching, and the writing of exercises, conclusions or other works in 15 hours.Total 90 hours.

Attendance and active participation in theoretical and practical classes. Taking advantage of tutoring. Previously advised subjects: Mathematical Methods I, II, III and IV.