Advanced Algebra and Calculus

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Credits
7.5
Types
Compulsory
Requirements
This subject has not requirements, but it has got previous capacities
Department
MAT
This course presents subjects of mathematics that extend or complement those introduced in the courses on Algebra and Calculus of the first semester.

Teachers

Person in charge

  • Jaume Franch Bullich ( )

Weekly hours

Theory
3
Problems
2
Laboratory
0
Guided learning
0
Autonomous learning
7.5

Competences

Technical Competences

Technical competencies

  • CE1 - Skillfully use mathematical concepts and methods that underlie the problems of science and data engineering.

Transversal Competences

Transversals

  • CT5 - Solvent use of information resources. Manage the acquisition, structuring, analysis and visualization of data and information in the field of specialty and critically evaluate the results of such management.
  • CT6 [Avaluable] - Autonomous Learning. Detect deficiencies in one's own knowledge and overcome them through critical reflection and the choice of the best action to extend this knowledge.

Basic

  • CB1 - That students have demonstrated to possess and understand knowledge in an area of ??study that starts from the base of general secondary education, and is usually found at a level that, although supported by advanced textbooks, also includes some aspects that imply Knowledge from the vanguard of their field of study.

Generic Technical Competences

Generic

  • CG2 - Choose and apply the most appropriate methods and techniques to a problem defined by data that represents a challenge for its volume, speed, variety or heterogeneity, including computer, mathematical, statistical and signal processing methods.
  • CG5 - To be able to draw on fundamental knowledge and sound work methodologies acquired during the studies to adapt to the new technological scenarios of the future.

Objectives

  1. Extension of knowledge of Algebra and Calculus.
    Related competences: CB1,
  2. Recognize and apply the concepts of Algebra and Calculus related to multidisciplinary problems.
    Related competences: CE1, CT5, CT6,
  3. Achieve a mastery of software that allows you to solve problems of greater complexity from the knowledge acquired.
    Related competences: CT5, CG2, CG5,

Contents

  1. Multiple integrals
    Riemann integral of functions of several variables. Rectangle; arbitrary domains; improper integrals. Fubini's Theorem. Iterated integrals. Normal domains. Change of variables theorem. Polar and spherical coordinates. Numerical methods. Quadrature formulas. Monte Carlo mehtod.
  2. Fourier series and Fourier transform
    Spaces of funcions. Sequences and series of funcions. Trigonometric and exponential Fourier series. Parity. Fourier transform. Properties: symmetries, shift, scaling, convolution, conservation of energy. Generalized functions. Dirac Delta. Functionals. Distributions.
  3. Quadratic forms and extrema
    Quadratic forms and symmetric matrices. Definite, indefinite, semidefinite. Diagonalization. Signature. Restriction to subspaces. Gradient, Jacobian, Hessian. Local extrema of functions of several variables. Critical points. Constrained extrema. Lagrange multipliers. Global extrema on compact sets.

Activities

Activity Evaluation act


Development of topic 1 of the course

Theory classes and Problems of the topic 1
Objectives: 1 2 3
Theory
12h
Problems
8h
Laboratory
0h
Guided learning
0h
Autonomous learning
30h

Development of topic 2 of the course

Theory and Problems classes on topic 2
Objectives: 1 2 3
Theory
21h
Problems
14h
Laboratory
0h
Guided learning
0h
Autonomous learning
52.5h

Development of topic 3 of the course

Theory and problem classes on topic 3
Objectives: 1 2 3
Contents:
Theory
12h
Problems
8h
Laboratory
0h
Guided learning
0h
Autonomous learning
30h

Teaching methodology

Theory classes will be in the form of master classes in which the contents of the subject will be explained and examples and illustrative problems will also be given.
In the problem classes, problems will be solved on the topics studied in theory.

Evaluation methodology

There will be two exams: a mid-course exam EP (which does not release a subject) and a final exam EF; in addition solved problems will have to be delivered and/or answer quizzes AC.

The grade of the subject in the ordinary call will be calculated as follows:

NF=max{0.60*EF+0.25*EP+0.15*AC; 0.75*EF+0.25*EP; EF}

There will be an extraordinary exam (EX). The grade in extraordinary call will be

Max(EX, NF).

Bibliography

Basic:

Complementary:

Previous capacities

Courses on Algebra and Calculus of the first term