Universidad Pública de Navarra - Nafarroako Univertsitate Publikoa

Module/Subject matter

Basic formation module / Mathematics

Up

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 

Up

Descriptors

Vector Calculus. Ordinary Differential Equations.

Up

General proficiencies

CG3: Knowledge of basic and technological subjects to have the ability to learn new methods and theories, and versatility to adapt to new situations

CG4: Analysis and synthesis ability

Up

Specific proficiencies

CB1: 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

Up

Learning outcomes

• • R1 Know the fundamentals of analytic and differential geometry
  • R2 Handle the fundamentals of differential calculus in several variables: gradient, divergence, rotational (curl) and Stokes¿s theorem
  • R3 Handle the basic concepts of Integral Calculus in several variables. Find lenghts of curves, and area and volume of solids in 3D.
  • R4 Know how to apply Calculus to Engineering problems.
  • R5 Understanding the notion of ordinary differential equation (ODEs). Know some basic analytic tools to solve the basic types of ODEs
  • R6 Know how to apply Partial Differential Equations: Wave equation and Heat equation

Up

Methodology

Up

Evaluation

There will be two written midterm exams which will cover the main parts the subject has been divided: Differential and Integral Vector Calculus and Ordinary Differential equations. These exams will count for 50% and 30% respectively of the total grade. There is also a homework assignment on selected topics of the first part of the course which count for the remainder 20%. A minimum grade of 3 is needed to pass the exam. Otherwise, the final grade will be the minimum between 4.9 and the average obtained from these exams.

If the student fails to pass the course 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, 70% 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 smartwatches 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)

Up

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. Applications

Up

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)

Up

Languages

English

Up

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)

Up