Public University of Navarre

Academic year: 2019/2020 | Previous academic years:  2018/2019  |  2017/2018 
Bachelor's degree in Computer Science at the Universidad Pública de Navarra
Course code: 250206 Subject title: MATHEMATICS II
Credits: 6 Type of subject: Basic Year: 1 Period: 2º S
PORTERO EGEA, LAURA (Resp)   [Mentoring ] ARRARAS VENTURA, ANDRÉS   [Mentoring ]

Partes de este texto:


Module/Subject matter

Basic training / Mathematics.



Elements of differential and integral calculus in several variables. Ordinary differential equations.


General proficiencies

General proficiencies that a student should acquire in this course:

  • G8 Knowledge of basic and tecnological subjects to have the ability to learn new methods and theories, and versatility to adapt to new situations
  • G9 Problem solving proficiency with personal initiative, decision making, creativity and critical reasoning. Ability to elaborate and communicate knowledge, abilities and skills in computer engineering
  • T1 Analysis and synthesis ability
  • T3 Oral and written communication
  • T4 Problem solving
  • T8 Self-learning


Specific proficiencies

Specific proficiencies that a student should acquire in this course:

  • FB1 Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on linear algebra, differential and integral calculus, numerical methods, numerical algorithms, statistics and optimization
  • FB3 Ability to understand and master the basic concepts of discrete matemathics, logic, algorithmics and computational complexity, and their applications to problem solving in engineering.


Learning outcomes

At the end of the course, the student is able to:

  • O1: Apply the basic elements of differential calculus in several variables: limits, continuity, differentiability.
  • O2: Formulate and solve unconstrained and constrained optimization problems.
  • O3: Apply the basic elements of integral calculus in several variables, e.g., to determine the length of a curve, the area of a surface, the volume of a solid, etc.
  • O4: Understand the basic elements of vector calculus: flux integral, gradient, divergence, curl, integral theorems.
  • O5: Recognize and solve some basic types of ordinary differential equations.



The following table shows the distribution of activities in the course:

Methodology - Activity  On-site hours  Off-site hours
 A-1: Theoretical lectures 45  
 A-2: Practical lectures 15  
 A-3: Self-study   80
 A-4: Exams and assessment 5  
 A-5: Tutoring 5  
 TOTAL 70 80


Relationship between formative activities and proficiencies/learning outcomes

Proficiency Activities
G8 A-1, A-2, A-3, A-5
G9 A-1, A-2, A-3, A-5
FB1 A-1, A-2, A-3, A-4, A-5
FB3 A-1, A-2, A-3, A-5
T1 A-1, A-2, A-3, A-5
T3  A-1, A-2, A-4
T4  A-2, A-4, A-5
T8 A-2, A-3






Continuous assessment is considered along the semester based on the following activities:

Learning outcome Assessment activity Weight (%) Resit assessment
O1, O2 Midterm exam A on lessons 1 and 2 35 Yes (final exam)
O3, O4 Midterm exam B on lessons 3 and 4 45 Yes (final exam) 
O5 Midterm exam C on lesson 5 20 (Minimum to be considered in the final mark: 3 out of 10) Yes (final exam)

In order to pass the subject, one of the following conditions must be fulfilled:

  • the mark of the midterm exam C is not less than 3 (out of 10) and the weighted mark of all three midterm exams is not less than 5 (out of 10);
  • the mark of the final exam covering the whole course (to be scheduled during the resit assessment period) is not less than 5 (out of 10). Only students who did not pass the course by continuous assessment can sit this exam.




  • Functions of several variables: limits, continuity, differentiation, Taylor series and graphics.
  • Multiple integration. Applications.
  • Vector calculus.
  • Ordinary differential equations.




  1. Functions, limits and continuity in Rn. Definition. Scalar and vector functions. Limits. Continuity.
  2. Differential calculus in Rn. Partial and directional derivatives. Gradient vector and Jacobian matrix. Differentiability. The chain rule. Higher-order derivatives. Hessian matrix. Taylor polynomials. Relative extrema. Constrained optimization: the theorem of Lagrange multipliers. Absolute extrema on closed bounded regions.
  3. Integral calculus in Rn. The Riemann integral. Elementary regions. Fubini's theorem. Change of variables. Polar, cylindrical and spherical coordinates.
  4. Vector calculus. Conservative fields. Potential function. Line and surface integrals. Circulation and flux. Divergence and curl. Green's, divergence and Stokes' theorems.
  5. Ordinary differential equations. Basic notions on differential equations. First-order ordinary differential equations. Existence and uniqueness of solution. Some elementary integration methods. Higher-order linear differential equations. Applications.



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

Basic bibliography:

  • R.A. Adams. Calculus: a complete course. Addison-Wesley.
  • E. Kreyszig. Advanced engineering mathematics. John Wiley & Sons.
  • J.E. Marsden, A.J. Tromba. Vector calculus. W.H. Freeman.
  • R.K. Nagle, E.B. Saff, A.D. Snider. Fundamentals of differential equations and boundary value problems. Addison-Wesley.

Additional bibliography:

  • M. Braun. Differential equations and their applications: an introduction to applied mathematics. Springer-Verlag.
  • R.E. Larson, R.P. Hostetler. Cálculo y geometría analítica. McGraw-Hill.
  • S.L. Salas, E. Hille, G.J. Etgen. Calculus: una y varias variables. Reverté.
  • D.G. Zill. Ecuaciones diferenciales con aplicaciones de modelado. Thomson.



Lecture room building at Arrosadia Campus.