Public University of Navarre

CastellanoEuskara | Academic year: 2019/2020 | Previous academic years:  2018/2019  |  2016/2017  |  2015/2016 
Bachelor's degree in Agricultural, Food and Rural Environment Engineering at the Universidad Pública de Navarra
Course code: 501206 Subject title: MATHEMATICS II
Credits: 6 Type of subject: Basic Year: 1 Period: 2º S
Department: Estadística, Informática y Matemáticas

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Module/Subject matter




Differential and integral calculus in several variables. Differential equations.


General proficiencies

  • CB1: Students are able to demonstrate they have acquired knowledge and understanding in a field of study based on the basic foundations gained within their general secondary education together with the support of advanced textbooks and aspects of the latest advances in the field.
  • CG2: Adequate knowledge of the physical problems, technologies, equipment, and water and energy supply systems, the limits imposed by budgetary factors and building regulations, the relationships between installations and/or buildings with farms, agro-food industries and spaces related to gardening and landscaping with their social and environmental surroundings, as well as the need to relate those surroundings from that environment with human needs and environmental protection.


Specific proficiencies

  • CE1: Ability to solve mathematical problems that may arise in engineering. Aptitude for applying knowledge towards: linear algebra, geometry, differential geometry, differential and integral calculus, differential equations and partial derivatives, numerical methods and numerical algorithmic methods, statistics, and optimization


Learning outcomes

R1- Operate with elementary functions (polynomial, rational, trigonometric, logarithmic, exponential...); know the fundamental properties of these functions and become familiar with the ideas of limit, continuity and derivability.
R2 - Work with analytical expressions and know how to simplify them.
R3 - Know geometric concepts related to functions of one and several variables: function graphs, curves and level surfaces, parameterized curves and surfaces.
R4- Identify and solve different integrals: single, double, triple, surface, line, their physical origin and their use inengineering problems.
R5- Know and apply fundamental theorems of calculus: GreenStokes and Divergence.
R6- Identify and solve simple differential equations: first order and second order with constant coefficients equations.
Handle a symbolic processor


ENAEE-1: Knowledge and understanding of the scientific and mathematical principles underlying their engineering branch.




The course consists of 6 credits, distributed in 4.5 theoretical and 1.5 practical credits. The whole-class theoretical lessons include the description of basic concepts and methods. The small-group practical sessions are devoted to the solution of problems and applications. Finally, teacher-assisted tutoring is available for students all along the semester in order to solve possible queries and questions arising in the study of the subject. The following table shows the distribution of the activities in the course:

Methodology - Activity  On-site hours  Off-site hours
A-1: Theoretical lessons 41  
A-2: Practical lessons 15  
A-3: Discussion 4  
A-6: Individual study    80
A-7: Exams and assessment 4  
A-8: Individual tutoring  10  
 TOTAL 70 80


Relationship between formative activities and proficiencies


Proficiency Formative activities
CB1 A-1, A-2, A-3, A-4
CG2 A-1, A-2, A-3, A-4
CE1 A-1, A-2, A-3, A-4







Learning outcome Assessment Weight(%) Resit assessment
R1, R2, R3, R4, R5. ENAEE-1 Long-answer exam questions 85%  Yes
R1, R2, R3, R4, R5. ENAEE-1 Individual work 15%  Yes


For assessment purposes, the course is divided into three parts:

  • Part A (related to lessons 1, 2 and 3) is worth 45%.
  • Part B (related to lessons 4 and 5) is worth 40%.
  • Individual coursework 15%.

In order to pass the subject the weighted mark of all three parts is not less than 5.



Vector functions of several variables. 
Ordinary and partial differential equations.
Integral Calculus of functions of several variables. Applications.



  1. Functions, limits and continuity in Rn. Basic concepts in scalar and vector functions of several variables. Limits. Continuity: definition and local and global properties.
  2. Differential calculus in Rn. Directional and partial derivatives. Jacobian matrix and gradient vector. Differentiability. Chain rule. Higher-order derivatives. Hessian matrix. Taylor series. Relative extrema. Constrained optimization: the theorem of Lagrange multipliers. Absolute extrema on closed bounded regions.
  3. Ordinary differential equations. Basic concepts of differential equations. First-order ordinary differential equations. Existence and uniqueness of solution. Some elementary integration methods. Higher-order linear differential equations. Systems of linear differential equations. Applications.
  4. Integral calculus in Rn. Riemann integral. Elementary regions. Fubini's theorem. Change of variable theorem. Polar, cylindrical and spherical coordinates. Line and surface integrals.
  5. Vector calculus. Vector fields. Divergence and curl. Line integrals. Conservative fields. Potential function. Flux integrals. Fundamental theorems of vector calculus.



Acceda a la bibliografía que su profesor ha solicitado a la Biblioteca.

  • R.A. Adams. Calculus: a complete course. Addison Wesley.
  • M. Braun. Differential equations and their applications: an introduction to applied mathematics. Springer-Verlag.
  • E. Kreyszig. Advanced engineering mathematics. John Wiley & Sons.
  • J.E. Marsden, A.J. Tromba. Vector calculus. W.H. Freeman.

Spanish textbooks:

  • R.E. Larson, R.P. Hostetler. Cálculo y geometría analítica. McGraw-Hill.
  • R.K. Nagle, E.B. Saff, Ecuaciones diferenciales y problemas con valores en la frontera. Pearson Educación.
  • S.L. Salas, E. Hille, G.J. Etgen. Calculus: una y varias variables. Reverté.
  • D.G. Zill. Ecuaciones diferenciales con aplicaciones de modelado. Thomson.



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