Public University of Navarre



Academic year: 2023/2024 | Previous academic years:  2022/2023  |  2021/2022  |  2020/2021  |  2019/2020 
Bachelor's degree in Industrial Engineering at the Universidad Pública de Navarra
Course code: 252101 Subject title: MATHEMATICS I
Credits: 6 Type of subject: Basic Year: 1 Period: 1º S
Department: Estadística, Informática y Matemáticas
Lecturers:
LIZASOAIN IRISO, INMACULADA (Resp)   [Mentoring ] ALBIAC ALESANCO, FERNANDO JOSÉ   [Mentoring ]
QUEMADA MAYORAL, CARLOS   [Mentoring ]

Partes de este texto:

 

Module/Subject matter

Basic training module / M11 Mathematics

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General proficiencies

  • CG3: Knowledge of basic and tecnological subjects to have the ability to learn new methods and theories, and versatility to adapt to new situations.
  • CG4: Problem solving proficiency with personal initiative, decision making, creativity and critical reasoning. Ability to elaborate and communicate knowledge, abilities and skills in industrial engineering.

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Specific proficiencies

  • CFB1: Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on linear algebra, geometry, differential geometry, calculus, differential equations, numerical methods, algorithms, statistics and optimization.

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Learning outcomes

At the end of this course, students will be able to

  • Understand and apply the definitions of vector space, linear system, matrix, determinant, matrix diagonalization and scalar product
  • Know analytic and differential geometry.
  • Know the concepts of real number, real function of one real variable, limit and derivative, and graph real functions of one real variable.
  • Know the basic concepts of Integral Calculus, find volumes, areas and lengths using integrals, and use numerical differentiation and integration techniques.
  • Apply Calculus to Engineering.

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Methodology

Methodology - Activity Attendance  Self-study
A-1 Lectures 45  
A-2 Practical clases 15  
A-3 Self-study   75
A-4 Exams, tests, and office hours 15  
Total 75 75

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Relationship between formative activities and proficiencies/learning outcomes

Proficiency Activities
CG3: Knowledge of basic and tecnological subjects to have the ability to learn new methods and theories, and versatility to adapt to new situations A-1 Lectures A-2 Practical clases A-3 Self-study A-4 Exams, tests, and office hours
CG4: Problem solving proficiency with personal initiative, decision making, creativity and critical reasoning. Ability to elaborate and communicate knowledge, abilities and skills in industrial engineering A-1 Lectures A-2 Practical clases A-3 Self-study A-4 Exams, tests, and office hours
CFB1: Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on linear algebra, geometry, differential geometry, calculus, differential equations, numerical methods, algorithms, statistics and optimization A-1 Lectures A-2 Practical clases A-3 Self-study A-4 Exams, tests, and office hours

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Languages

English, Spanish and Basque

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Evaluation

 

Learning
outcome
Assessment
activity
Weight (%) It allows
test resit
Minimum
required grade
O1, O2, 03, O4, O5 Recurrent written quizzes that assess the daily work of the students 30% YES NO
O1, O2, 03, O4, O5 Final exam (long answer questions and practical cases) 70% YES To pass the course, the minimum required grade in this part is 4 out of 5.
         
         

This subject has two parts: Calculus (1/3 of the subject matter) and Linear Algebra (the remaining 2/3 of the subject matter). The weight of each part in the final grade is in accord with that distribution of credit hours. 

A Calculus midterm exam will take place right after finishing Unit 2. This test may leave out the Calculus part from the First Final Exam.

If a student does not fulfill the minimum requirements to pass the subject in some of the evaluation activities, the maximum grade that he or she will obtain is 4.9 out of then (failing grade).

 

 

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Contents

Vector Spaces. Matrices and determinants. Linear systems. Diagonalization of matrices.
Analytic and differential geometry. Euclidean geometry equations.
Functions of a real variable. Limits. Introduction to Differential Calculus. Differentiation. Applications.
Integration techniques. Introduction to Integral Calculus of functions of one real variable. Applications.

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Agenda

Lesson 1. Calculus in one single variable

Differentiation of functions: review.

Taylor polynomials.

Approximation of functions by polynomials.

Taylor theorem with Lagrange's remainder.

Lesson 2. Integration of functions of one variable.

The Riemann integral: definition and properties.
Mean value theorem for integrals.
Fundamental Theorems of Calculus.
Integration techniques. Integration by parts.
Change of variable.
Improper integrals.

Lesson 3. Matrices and linear systems

Systems of linear equations. Matrix notation.

Matrix multiplication: block matrices and block multiplication.

The inverse of a matrix.

Row reduction and echelon forms: The Gaussian row reduction algorithm.

Gaussian elimination.

Elementary matrices.

Solution sets of linear systems.

Determinants.

 

Lesson 4. Vector Spaces
Definitions and examples.
Linear combinations and linear independence.
Subspaces.
Bases and dimension.

The  change of basis matrix in a vector space.
Row space and column space of a matrix.
Solutions of linear systems.

 

Lesson 5. Linear Transformations.
Definitions and examples. Matrix associated to a linear transformation.
Operations with linear transformations.

Change of bases and linear transformations.

The null space of a linear transformation.

Kernel, image and isomorphisms. Rank+Nullity theorem.

 

Lesson 6. Diagonalization of linear transformations and matrices.

Linear transformations from a vector space into itself.

Eigenvalues and eigenvectors.
Diagonalization of matrices.

 

Lesson 7. Orthogonality.
Euclidean spaces. Scalar product.
Orthogonal projections. Gram-Schmidt method.
Orthogonal diagonalization of symmetric matrices.

 

Lesson 8. Least-squares approximations.

Approximate solutions of inconsistent systems.

Optimal least-squares solutions.

Normal equations.

 

 

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Bibliography

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


  • D. C. Lay, Linear Algebra and its applications, Pearson Education, 2006.
  • J. Hefferon, Linear Algebra. Available for free download from http//joshua.smcvt.edu/linearalgebra
  • G. Strang, Linear Algebra and its applications, 4th Edition, Thompson.
  • Larson and Edwards, Calculus of a single variable, Brooks/Cole.
  • J. Stewart, Calculus: Early Trascendentals, Brooks/Cole, 7th Edition 2012.
  • G. Strang, Calculus, Wellesley-Cambridge Press.
  • D. R. Guichard, Calculus. Available for free download from http://whitman.edu/mathematics/california_calculus/
  • R. Adams and C. Essex, Calculus, a complete course, 9th Edition, Pearson Education, 2018.

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