Public University of Navarre



Castellano | Academic year: 2014/2015
Bachelor's degree in Mechanical Design Engineering at the Universidad Pública de Navarra
Course code: 251101 Subject title: MATHEMATICS I
Credits: 6 Type of subject: Basic Year: 1 Period: 1º S
Department: Mathematics and Computer Engineering
Lecturers:
BELLOSO EZCURRA, JOSE JAVIER   [Mentoring ]

Partes de este texto:

 

Contents

Real functions in one variable. Limit. Continuity. Differentiation. Extrema and optimization. Taylor polynomial. Integration in one variable. 

 

Linear system of equations. Vector spaces. Orthogonality. Determinants. Eigenvalues and eigenvectors. 

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Descriptors

Single Variable Calculus. Algebra.

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General proficiencies

  • G1: Self-learning ability
  • G2: Problem solving proficiency with personal initiative, decision making, creativity and critical reasoning. Ability to elaborate and communicate knowledge, abilities and skills in engineering
  • G4: Knowledge of basic and tecnological subjects to have the ability to learn new methods and theories, and versatility to adapt to new situations
  • G5: Analysis and synthesis ability

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Specific proficiencies

E1: Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on linear algebra, differential geometry, calculus, differential equations, numerical methods, algorithms, statistics and optimization

 

 

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Methodology

Methodology - Activity

Attendance

 Self-study

A-1 Lectures

44

 

A-2 Practical clases

16

 

A-3 Debates, group study, etc

 

 

A-4 Assignments

 

 5

A-5 Readings

 

 

A-6 Self-study

 

 70

A-7 Exam, evaluation tests

3

 

A-8 Office hours

12

 

Total

75

75

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Evaluation

There are two alternatives for passing this subject:

 

1 Ordinary path:

  • Two written exams, which will evaluate the knowledge of Calculus and Algebra separately. These exams will be counted 40% each of the total grade.
  • A homework assignment on selected topics in Algebra,  which will count the remaining 20%  

The student must achieve a score of 5 (over 10) or above and attain at least a score of 4 (again over 10) in the exams and  assignment homework previously mentioned.

 

2 Extraordinary examination call:  it is also available for students who fail to pass the exam in the ordinary path

  • There is only one written exam. The exam will be divided into two parts, the first part is focused on Calculus and the second part, in Algebra. These parts will count 40% and 60% respectively in the total grade. In both parts, the student must achieve a score of 4 (over 10) or above to pass the subject. 

 

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Agenda

Lesson 0. Introduction 
Number sets. Complex numbers. Sets in R.  Absolute value function. Inequalities

 

 

First part 

 

Lesson 1. Functions in one variable, limits and continuity.  

Basic notions on real functions in one variable. Continuity: definition and local properties . Weierstrass, Bolano and Intermediate value Theorems

Lesson 2: Differential calculus

Derivative of a function at a point: definitions, interpretation and first properties. Derivative function. Derivatives of higher degree. Chain's rule. Rolle and Mean Value Theorems. Applications: extrema, L'Hopital rule. Finding functions zeros. Taylor and Maclaurin polynomials. 

 

Lesson 3: Integral calculus 

Riemann integral: definition and properties. Integral mean value theorem. Fundamental Calculus Theorem. Barrow rule. Integration by parts. Change of variables.  

 

Second part

 

Lesson 1: Linear systems of equations 

Linear systems of equations. Gauss elimination. Gauss elimination with pivoting. Gauss-Jordan method. Matrix form of a linear system. Matrix definition and first properties. Matrix product. Inverse matix. Rank. Rouche-Frobenius theorem

 

Lesson 2: Vector spaces in Rn  

Null and Column space of a matrix. Vector subspace.  Linear dependence and independence. Basis. Coordinates. Dimension of a subspace

 

Lesson 3: Inner product and Euclidean spaces 

Orthonormal basis. Gram-Schmidt algorithm. Orthogonal matrix. QR decomposition. Orthonormal basis. Least square approximation. Pseudoinverse.

 

Lesson 4: Determinants

Definition  and main properties. Cramer rule. 

 

Lesson 5: Eigenvalues and eigenvectors

Definition. Matrix characteristic polynomial. Diagonalisation of a matrix. Eigenvalues and eigenvectors for symmetric matrices. Quadratic form. Singular value decomposition.

 

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Bibliography

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


  • G.L. Bradley, K.J. Smith: Cálculo de una variable. Prentice Hall.
  • Cálculo I: Teoría y problemas de Análisis Matemático en una variable.  CLAGSA.
  • S.L. Salas, E. Hille and Etgen: Calculus: one variable. Wiley
  • J. B. Fraleigh, A. Beauregard,  Linear algebra Pearson
  • D. H. Griffel, Linear Algebra and its applications (Dos volúmenes), Ellis Horwood Ltd. 
  • S. Lang, Introduction to linear algebra. Springer
  • L. Merino y E. Santos, Álgebra lineal con métodos elementales, Thompson.
  • G. Strang, Linear Algebra and its applications,  TBS

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Languages

English

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