Universidad Pública de Navarra - Nafarroako Univertsitate Publikoa

Module/Subject matter

Basic formation module / Mathematics

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Contents

Real and vector functions in several variables. Differentiability. Extrema and inflection points. Constrained extrema and Lagrange multipliers. Taylor polynomial

Integration in several variables, change of variables, non-cartesian coordinates, scalar and vector line integrals, fluxes. Fundamental Vector Calculus Theorems 

Ordinary differential equations. Initial value problems. Linear differential equations. Basic solution techniques. Applications 

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Descriptors

Differential and Integral vector calculus. Ordinary Differential Equations.

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General proficiencies

• G1: Self-learning ability
  
• G2: Problem solving proficiency with personal initiative, decision making, creativity and critical reasoning. Ability to elaborate and communicate knowledge, abilities and skills in engineering
• G4: Knowledge of basic and tecnological subjects to have the ability to learn new methods and theories, and versatility to adapt to new situations
• G5: Analysis and synthesis ability

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Specific proficiencies

E1: Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on linear algebra, differential geometry, calculus, differential equations, numerical methods, algorithms, statistics and optimization

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Learning outcomes

• Handle the fundamentals of differential calculus in several variables.
• State and solve problems of finding free and constrained extrema.
• Handle the basic concepts of Integral Calculus in several variables. Find lenghts of curves, and area and volume of solids in 3D. 
• Become familiar with the Fundamentals Theorems on Vector Calculus: Green, Stokes and Divergence theorem.
• Acquire starting notions on potential theory.
• Know how to apply the afore mentioned results to Engineering problems. 
• Understand the notion of ordinary differential equation (ODEs).
• Know some basic analytic tools to solve selected ODEs 

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Methodology

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Evaluation

There will be three written midterm exams which will cover the main parts the subject has been divided: Differential Vector Calculus, Integral Vector Calculus and Differential equations. These exams will count for 30%, 40% and 30% respectively of the total grade. A minimum grade of 3.5 is needed to pass the exam. Otherwise, the final grade will be the minimun between 4.9 and the average obtained from these exams.

If the student fails to pass the subject following the ordinary path described above, he/she can apply for an extraordinary call examination which consists in a unique final exam. The exam will be structured as this subject: It will be divided into three parts (Differential, Integral calculus and Differential equations) which will count for the same ratios, 30%, 40% and 30%, in the final grade.

Students can use for the exams, notes and any book he/she considers appropriate. Programmable calculators and electronic devices such as laptops, tablets, and smartwhatchs are banned.

(Updated information on timetable and venue for the exams can be found at  http://www.unavarra.es/ets-industrialesytelecos/estudios/grado/grado-en-ingenieria-en-disenio-mecanico-campus-de-tudela/periodos-de-evaluacion?submenu=yes)

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Agenda

1st Part: Continuity and differentiability 

Lesson 1: Several variables functions 

Sets in  Rⁿ. Real and vector functions in several variables. Limit. Continuity

Lesson 2: Differential calculus in Rⁿ

Partial derivatives. Gradient. Chain rule. Inverse and implicit function. Taylor polynomial. Extrema and saddle points. Constrained extrema and Lagrange multipliers  

2nd Part: Integral vector calculus

Lesson 3: Integral in several variables

Double integral. Triple integral. Non-cartesian coordinates and change of variables

Lesson 4. Path integrals. Flux. Fundamental vector calculus

Scalar and vector path integrals. Flux. Green, Stokes and Divergence Theorem. Potential theory.

3rd Part: Ordinary differential equations

Lesson 5: Ordinary differential equations

Definition and first properties. Equations of first order. Initial value problem. Differential linear equations of order n. Contour problems. Applications

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Bibliography

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

Basic bibliography:

1. S.L.  Salas, E. Hille, G. J. Etgen: Calculus. Reverté
2. V. Domínguez, Apuntes de Cálculo Vectorial, disponible en miaularario
3. D. G. Zill, A First Course in Differential Equations with Modeling Applications. Cengage Learning.

Complementary Bibliography

1. J. E. Marsden y A. J. Tromba: Vector Calculus. Macmillan Higher Education
2. M. D. Weir: Thomas calculus. Pearson-Addison Wesley.    
3. R.K. Nagle, E.B. Saff, A. David Fundamentals of differential equations. Pearson

( Library Catalogue can be consulted at  https://biblioteca.unavarra.es/abnetopac/abnetcl.cgi/O7164/ID7e647614?ACC=101)

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Languages

English

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Location

Classroom 

(Updated information on timetables and classrooms can be found at  http://www.unavarra.es/ets-industrialesytelecos/estudios/grado/grado-en-ingenieria-en-disenio-mecanico-campus-de-tudela/horarios?submenu=yes)

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