Universidad Pública de Navarra - Nafarroako Univertsitate Publikoa
Public University of Navarre
| Course code: 252101 | Subject title: MATHEMATICS I | ||||
| Credits: 6 | Type of subject: Basic | Year: 1 | Period: 1º S | ||
| Department: Estadística, Informática y Matemáticas | |||||
| Lecturers: | |||||
| ALBIAC ALESANCO, FERNANDO JOSÉ (Resp) [Mentoring ] | |||||
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.
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.
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.
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.
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 |
Evaluation
| Learning outcome |
Assessment activity |
Weight (%) | It allows test resit |
Minimum required grade |
|---|---|---|---|---|
| O1, O2, 03, O4, O5 | 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 10. |
This subject has two parts: Linear Algebra (2/3 of the program) and Calculus (the remaining 1/3 of the program). The weight of each part in the final grade is in accord with that distribution of credit hours.
There will be a Linear Algebra test right after Unit 5 and a Calculus test before finishing Unit 7.
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).
Agenda
Unit 0. Matrices and linear systems
Unit 1. 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.
Unit 2. Linear Transformations.
Definitions and examples. Matrix associated to a linear transformation.
Operations with linear transformations.
Change of bases and linear transformations.
The null space and the image of of a linear transformation.
Rank+Nullity theorem.
Unit 3. Diagonalization of linear transformations and matrices.
Linear transformations from a vector space into itself.
Eigenvalues and eigenvectors.
Diagonalization of matrices.
Unit 4. Orthogonality.
Euclidean spaces. Scalar product.
Orthogonal projections. Gram-Schmidt method.
Orthogonal diagonalization of symmetric matrices.
Unit 5. Least-squares approximations.
Approximate solutions of inconsistent systems.
Optimal least-squares solutions.
Normal equations.
Unit 6. Calculus in one single variable
Differentiation of functions: review.
Taylor polynomials.
Approximation of functions by polynomials.
Taylor theorem with Lagrange's remainder.
Unit 7. 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.
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.