Public University of Navarre



Castellano | Euskara | Academic year: 2015/2016 | Previous academic years:  2014/2015 
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: Mathematics and Computer Engineering
Lecturers:
ROLDAN MARRODAN, ANGEL TEODORO (Resp)   [Mentoring ]

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Module/Subject matter

Mathematics

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Contents

The contents of this course provides students with the basic principles and concepts of differential and integral calculus in several variables and differential equations with a focus on its application in Agricultural Engineering and Rural Environment.

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Descriptors

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

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General proficiencies

General proficiencies that a student should acquire in this course: CT6, CT7.

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Specific proficiencies

Specific proficiencies that a student should acquire in this course: CE1.

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Learning outcomes

- Operate with elementary functions (polynomial, rational, trigonometric, logarithmic, exponential, ...); know its fundamental properties and become familiar with the ideas of limit, continuity and derivability
- Work with analytical expressions and know how to simplify them.
- Know geometric concepts related to the functions of one and several variables: function graphs, curves and level surfaces, parameterized curves and surfaces.
- Identify and solve different integrals: single, double, triple, surface, line, their physical origin and their use in engineering problems.
- Know and apply fundamental theorems of calculus: Green, Stokes and Divergence.

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Methodology

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: Individual study    80
 A-4: Exams and assessment 4  
 A-5: Individual tutoring  10  
     
 TOTAL 70 80

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Languages

English

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Evaluation

The contents of the course are assessed by means of two written exams. There is a continuous assessment which involves the 35% of the final mark. The rest of the contents of the subject will be assessed in a final exam, and will involve the 65% of the final mark.

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Agenda

  • Functions, limits and continuity in Rn. Basic concepts in scalar and vector functions of several variables. Limits. Continuity: definition and local and global properties.
  • 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.
  • Integral calculus in Rn. Riemann integral. Elementary regions. Fubini's theorem. Change of variable theorem. Polar, cylindrical and spherical coordinates. Line and surface integrals.
  • Vector calculus. Vector fields. Divergence and curl. Line integrals. Conservative fields. Potential function. Flux integrals. Fundamental theorems of vector calculus.
  • 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.

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Bibliography

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


  • 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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Location

Arrosadia Campus

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