Universidad Pública de Navarra - Nafarroako Univertsitate Publikoa

Contents

• Functions of several variables: limits, continuity, differentiation and graphics.
• Taylor series in several variables.
• Multiple integration. Applications.
• Vector calculus.
• Ordinary 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: G8, G9, T1, T3, T4, T8

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Specific proficiencies

Specific proficiencies that a student should acquire in this course: FB1

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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

The contents of the course are assessed by means of two written exams. There is a mid-term exam covering the first third of the course syllabus, which involves the 33% of the final mark. The rest of the contents of the subject will be assessed in a final exam, and will involve the 67% of the final mark.

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Agenda

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. Integral calculus in Rⁿ. Riemann integral. Measurable sets. Elementary regions. Fubini's theorem. Change of variable theorem. Polar, cylindrical and spherical coordinates. Line and surface integrals.
4. Vector calculus. Vector fields. Divergence and curl. Line integrals. Conservative fields. Potential function. Green's theorem. Flux integrals. Stokes and divergence theorems. Circulation and flux.
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. 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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Languages

English

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Location

Arrosadia Campus

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