Public University of Navarre



Castellano | Euskara | Academic year: 2016/2017 | Previous academic years:  2015/2016  |  2014/2015 
Bachelor's degree in Agricultural, Food and Rural Environment Engineering at the Universidad Pública de Navarra
Course code: 501101 Subject title: MATHEMATICS I
Credits: 6 Type of subject: Basic Year: 1 Period: 1º S
Department: Mathematics and Computer Engineering
Lecturers:
YANGUAS SAYAS, PATRICIA   [Mentoring ]

Partes de este texto:

 

Module/Subject matter

Mathematics/ Mathematics I.

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Contents

Funciones, límites y continuidad. Conceptos básicos sobre funciones escalares o vectoriales de una o varias variables reales. Funciones elementales. Conjuntos de nivel. Límites. Continuidad de una función en un punto. Propiedades de funciones continuas.

Espacios vectoriales sobre R: Subespacios. Base y dimensión

Espacio euclídeo: producto escalar y norma euclídea, bases ortonormales, ortogonalización de Gran-Schmidt.

Diagonalización de matrices: valores y vectores propios. Subespacios fundamentales. Aproximación por mínimos cuadrados.

Matrices: matriz inversa, sistemas lineales, Teorema de Rouché-Frobenius. Determinantes

Cálculo vectorial: campos vectoriales en R2 y R3. Divergencia y rotacional, integrales de línea, campos conservativos, función potencial, teorema de Green, integrales de flujo, teorema de Stokes, teorema de divergencia. Circulación y flujo.

 

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Descriptors

Differential and integral calculus, linear algebra.

Keywords: Applied mathematics, derivative, integral, matrix, vector.

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General proficiencies

  • CB1: Students are able to demonstrate they have acquired knowledge and understanding in a field of study based on the basic foundations gained within their general secondary education together with the support of advanced textbooks and aspects of the latest advances in the field.
  • CB5: Students can develop the necessary learning skills to undertake further studies with a high degree of autonomy.

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Specific proficiencies

  • CE1: Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on: linear algebra; geometry; differential geometry; differential and integral calculus; differential equations and partial differential equations; numerical methods, numerical algorithms, statistics and optimization.
  • CG2: Adequate knowledge of the physical problems, technologies, equipment, and water and energy supply systems, the limits imposed by budgetary factors and building regulations, the relationships between installations and/or buildings with farms, agro-food industries and spaces related to gardening and landscaping with their social and environmental surroundings, as well as the need to relate those surroundings from that environment with human needs and environmental protection.

 

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Learning outcomes

  • (R1) Work with elementary functions (polynomials, rational, trigonometric, logarithms, exponentials,...), know their elementary properties and have an idea of the concepts of limit, continuity and differentiablility.
  • (R2) Work with analytic expressions and simplify and/or find apropriate bounds for them.
  • (R3) Solve approximation problems: Taylor, least squares, interpolation,...
  • (R4) Identify and compute the elementary forms of integrals in one variable; work with their physical and engineering applications and calculate volumes and areas.
  • (R5) Know the theoretical foundations and the direct algorithms for the resolution of algebraic linear systems.
  • (R6) Dominate the diagonalization of matrices: eigenvalues and eigenvectors.


    RESULTADOS DE APRENDIZAJE ENAEE

    ENAEE-1: Conocimiento y compresión de los principios científicos y matemáticos que subyacen a su rama de ingeniería.

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Methodology

Methodology - Activity Attendance Self-study
A-1 Lectures 44  
A-2 Practical classes 16  
A-6 Self-study   75
A-7 Exams, evaluation tests 5  
A-8 Office hours 10  
Total 75 75

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Languages

English, Spanish and Basque.

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Evaluation

 

 Learning outcome   Evaluation system  Weight (%)   Possibility of resit 
 (R1)-(R4)  Continuous evaluation   50  YES
 (R5)-(R6)  Continuous evaluation  50  YES

Continuous evaluation:

There are two partial exams: the first one corresponds to Chapters 1 and 2 and the second one accounts for the rest. To pass the subject by averaging the grades in the two partial exams it is necessary to obtain a minimum of 3 points out of 10 in each partial exam.

The active participation during lessons will be positively evaluated.

Resit evaluation:

It is a written exam that corresponds to the whole subject. It will represent a 100 % of the final grade.

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Agenda

1. Real-valued functions of a real variable

Real numbers. Limits and continuity. Differentiation. Applications: Taylor polynomial, optimization,... Calculus of zeros of a function.

2. Integrals of real-valued functions of a real variable

Definition and properties. The fundamental theorem of calculus. Elementary integration methods. Applications.

3. Vectors and matrices

Linear combination of vectors, linear independence, bases, dimension and coordinates. Matrices: rank, determinant and inverse matrix. Linear systems. Applications.

Vector length and orthogonality. Orthogonal projection. Least squares method. Construction of orthogonal bases. Applications.

4. Matrix diagonalization

Eigenvalues and eigenvectors. Characteristic polynomial. Eigenspaces. Algebraic and geometric multiplicity. Diagonalizable matrices. 

Polynomial functions of matrices. Quadratic forms. Diagonalization and classification of quadratic forms.

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Bibliography

Access the bibliography that your professor has requested from the Library.


BASIC BIBLIOGRAPHY:

Calculus: A Complete Course (7th edition), R. A. Adams and C. Essex, Pearson Education, Canada, 2009.

Introduction to Linear Algebra (2nd edition), S. Lang, Undergraduate Texts in Mathematics, Springer, New York, 1986.

Calculus With Analytic Geometry (8th edition), R. Larson, R.P. Hostetler, Houghton Mifflin Company, 2005.

Calculus: One and several variables (10th edition), S. Salas, E. Hille, G. Etgen, John Wiley & Sons Inc., United States of America, 2007.

Linear Algebra and Its Applications (4th edition), G. Strang, Cengage Learning, United Kingdom, 2006.

Calculus (3rd edition), M.J. Strauss, G.L. Bradley and K.J. Smith, Prentice Hall, 2002.

SUPPLEMENTARY BIBLIOGRAPHY:

Linear Algebra and Its Applications: A first course, D.H. Griffel, E. Horwood, 1989.

Linear Algebra and Its Applications (5th edition), D.C. Lay, S.R. Lay and J.J. McDonald, Pearson Education Limited, Edimburgh Gate, 2016.

Mathematics for Life Sciences, C. Neuhauser, Prentice Hall PTR, 2001.

Linear Algebra with Applications (7th edition), W.K. Nicholson, McGraw-Hill Ryerson Ltd., Toronto, 2013.

A course in Linear Algebra with Applications (2nd edition), D.J.S. Robinson, World Scientific Pub Co Inc., Singapore, 2006.

Calculus (3rd edition), J. Rogawsky and C. Adams, W. H. Freeman Publishers, 2015.

Calculus With Analytic Geometry (2nd edition), G.F. Simmons, McGraw-Hill Education, 1995.

Thomas' Calculus: Early Trascendentals (13th edition), G.B. Thomas Jr., M.D. Weir, J.R. Hass, Pearson Addison Wesley Ed., Boston, 2013.

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Location

Lecture rooms building.

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