Universidad Pública de Navarra - Nafarroako Univertsitate Publikoa

Module/Subject matter

Basic training / M11 Mathematics.

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Contents

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

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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:

• 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 engineering.

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Specific proficiencies

Specific proficiencies that a student should acquire in this course:

• 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.

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Learning outcomes

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

O1. Apply the basic elements of differential calculus on several variables: gradient, divergence, curl, Stokes theorems.
O2. 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,... using integrals, and use numerical differentiation and integration techniques.
O3. Apply Calculus to Engineering.
O4. Understand the concept of differential equation, and solve basic ordinary differential equations.
O5. Apply partial differential equations: wave equation and heat equation.

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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:
 

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Evaluation

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

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

• Part A (related to lessons 1 and 2) is worth 35%.
• Part B (related to lessons 3 and 4) is worth 45%.
• Part C (related to lesson 5) is worth 20%.

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

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

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Agenda

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

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Bibliography

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, Ecuaciones diferenciales y problemas con valores en la frontera. Pearson Educación.

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.

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Languages

English, Spanish and Basque.

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Location

Lecture room building at Arrosadia Campus.

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