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 tests that assess the daily work of the students | 33.3% | YES | NO |
| O1, O2, 03, O4, O5 | Final exam (long answer questions and practical cases) | 66.7% | YES | To pass the course, the minimum required grade in this part is 4 out of 10. |
There will be a test on the Calculus part right after finishing Unit 2. Those students who get a passing grade in that part will not have to redo it in the (ordinary) 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 ten (failing grade).
Agenda
FIRST PART: CALCULUS
Unit 1. Calculus in one single variable
Differentiation of functions: review.
Taylor polynomials.
Approximation of functions by polynomials.
Taylor theorem with Lagrange's remainder.
Unit 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.
SECOND PART: LINEAR ALGEBRA
Unit 3. Linear systems and Matrices.
Gaussian elimination.
Linear systems in matrix form.
Matrix algebra. The inverse of a matrix.
Row reduction and échelon forms.
Solution sets of linear systems. The Rouché-Fröbenius Theorem.
Determinants.
Unit 4. The vector Space Rn
Definitions and examples.
Linear combinations and linear independence.
Subspaces.
Bases and dimension.
The change of basis matrix in a vector subspace.
Row space and column space of a matrix.
The structure of the solution set of a linear system.
Unit 5. Orthogonality and Least squares.
Rn as Euclidean space. Dot product, norm and length of a vector.
Orthogonal projections. Gram-Schmidt method.
Orthogonal matrices.
Approximate solutions of inconsistent systems.
Optimal least-squares solutions.
Normal equations. Applications to best-fitting curves.
Unit 6. Linear Transformations.
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 7. Diagonalization of matrices.
Eigenvalues and eigenvectors.
Diagonalization of matrices.
Orthogonal diagonalization of symmetric matrices.
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.