Advanced Algebra and Calculus

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Credits
7.5
Types
Compulsory
Requirements
This subject has not requirements
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

  • Jordi Quer Bosor ( )

Others

  • Fernando Martínez Sáez ( )

Weekly hours

Theory
3
Problems
0
Laboratory
2
Guided learning
0.467
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 - 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. Ampliació del coneixements d'Àlgebra y Càlcul.
    Related competences: CB1,
  2. Reconèixer i aplicar els conceptes d'Àlgebra i Càlcul relacionats amb problemes multidisciplinars.
    Related competences: CE1, CT5, CT6,
  3. Aconseguir un domini del programari que permeti resoldre problemes d'una complexitat superior a partir dels coneixements adquirits.
    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
    Trigonometric and exponential Fourier series. Parity. Fourier transform. Properties: symmetries, shift, scaling, convolution, conservation of energy. Discrete-time transform. Discrete transform. Analogies and applications. Other transforms.
  3. Quadratic forms and extrema
    Quadratic forms and symmetric matrices. Definite, indefinite, semidefinite. Diagonalization. Signature. Extrema of functions of several variables. Critical points. Hessian matrix. Constrained extrema. Lagrange multipliers.
  4. Affine Geometry
    Affine space. Affine frames. Affine and barycentric coordinates. Affine varieties and their equations. Parallelism. Affine maps. Fixed points and invariant subvarieties. Translation, homothety, projection, symmetry.
  5. Euclidean Geometry
    Euclidean affine space. Orientation. Distance. Orthogonality. Volume. Orthogonal projection. Linear and affine isometries. Rigid motions. Classification and characterization of motion in dimensions 2 and 3. Spectral theorem in arbitrary dimension.
  6. Approximation and Least Squares
    Interpolation: interpolating polynomial; splines; cubic splines. Fitting: regression; least squares; principal components.

Activities

Activity Evaluation act


Desenvolupament del tema 1 de l'assignatura

Classes de Teoria i Laboratori del tema1
Objectives: 1 2 3
Contents:
Theory
9h
Problems
0h
Laboratory
6h
Guided learning
0h
Autonomous learning
20h

Desenvolupament del tema 2 de l'assignatura

Classes de Teoria i Laboratori del tema1
Objectives: 1 2 3
Theory
4h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
9h

Desenvolupament del tema 3 de l'assignatura

Classes de Teoria i Laboratori del tema3
Objectives: 1 2 3
Contents:
Theory
9h
Problems
0h
Laboratory
6h
Guided learning
0h
Autonomous learning
23h

Desenvolupament del tema 4 de l'assignatura

Classes de Teoria i Laboratori del tema 4
Objectives: 1 2 3
Theory
9h
Problems
0h
Laboratory
6h
Guided learning
0h
Autonomous learning
23.5h

Desenvolupament del tema 5 de l'assignatura

Classes de Teoria i Laboratori del tema 5
Objectives: 1 2 3
Theory
14h
Problems
0h
Laboratory
10h
Guided learning
0h
Autonomous learning
37h

Teaching methodology

Per les classes de teoria s'impartiran classes magistrals en les que també es faran exemples i problemes il·lustratius.
Les classes de laboratori es dedicaran al desenvolupament de programes en python sobre els temes de l'assignatura.

Evaluation methodology

Hi haurà dos exàmens: un parcial a mig curs (que no allibera matèria) i un examen final; a més s'hauran d'entregar problemes resolts.

La nota de l'assignatura en la convocatòria ordinària es calcularà de la manera següent:

MAX(0.6*F+0.3*P;0.9*F)+0.1*L

La nota en la convocatòria extraordinària serà la nota de l'examen.

Bibliography

Basic:

Complementary: