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General information

Subject type: Basic

Coordinator: Julián Horrillo Tello

Trimester: Second term

Credits: 6

Teaching staff: 

Carlos Paul Recarens

Skills

Basic skills

  • B5_That students have developed those learning skills necessary to undertake further studies with a high degree of autonomy

     

Specific skills

  • E1_Training for the resolution of mathematical problems that may arise in engineering. Train to apply knowledge about: linear algebra; geometry; differential geometry; differential and integral calculus; differential equations and partial derivatives; numerical methods; numerical algorithm; statistics and optimization

Transversal competences

  • T2_That students have the ability to work as members of an interdisciplinary team either as one more member, or performing management tasks, in order to contribute to developing projects with pragmatism and a sense of responsibility, making commitments taking into account available resources

     

Description

The subject provides a second mathematical level to students, completing the analysis of one variable with the integral and the analysis of functions in several variables.

New vector concepts related to derivation and integration with practical applications in electrical and mechanical engineering are introduced.

At the end of the course, the student must be able to:

  1. Calculate integrals by basic methods
  2. Calculate areas and volumes using Integral Calculation resources
  3. Solve the differentiability in several variables.
  4. Solve elementary situations of differential geometry
  5. Calculate the ends of the graph of functions of several variables
  6. Familiarize yourself with the vector notation of fields.

 

Learning outcomes

  1. Correctly apply the fundamental concepts of differential and integral calculus, linear algebra and geometry, developing the ability to apply them to engineering problems. (CE1)
  2. Appropriate use of differential equations in modeling and problem solving in engineering. (CE1)
  3. Ability to use the necessary mathematical tools in solving analytical and numerical problems. (CE1)
  4. Use the fundamental concepts of non-deterministic analysis and statistics, in engineering problems. (CE1)
  5. Analyze and critique the results of engineering problems (CE1)
  6. Solve problems in a work team.

Working methodology

It is based on the presentation in class of the theoretical concepts and the solution of problems that to a large extent, it is necessary that the student solves.

In the theoretical sessions the students have all the necessary information to follow the explanations of the professor and / or to study them of autonomous form by means of the notes, so much of theory as of multitude of problems solved and others that have to solve the students. 

The theoretical contents are illustrated in an applied and practical way, within the context of mathematics as an abstract conceptual subject.

The organization of the subject consists of differentiating the class activities and the activities of the students, these are in the eCampus of the subject defined and classified in the timetable.

 

 

Contents

1. Integral

1.1 Concept of Antiderivative
1.2 Areas and distances
1.3 Integral Defined
1.4 Fundamental theorem of calculus
1.5 Indefinite Integrals
1.6 The variable change rule

2. Applications of Integration I

2.1 Areas between curves
2.2 Volumes
2.3 Volumes by cylinders
2.4 Work
2.5 Average value of a function

3. Integration Techniques

3.1 Integration by parts
3.2 Trigonometric integrals
3.3 Trigonometric substitution
3.4 Integration of rational functions by means of partial fractions
3.5 Integrals using Integral Tables
3.6 Improper Integrals

4. Applications of Integration II

4.1 Arc length
4.2 Area of ​​a surface of revolution
4.3 Moment of center of mass
4.4 Pappus' theorem
4.5 Concepts of probability
4.6 Concept of differential equation

5. Vectors and Geometry in Space

5.1 Three-dimensional coordinate systems
5.2 Scalar Product
5.3 Vector Product
5.4 Vector Functions

6. Partial Derivatives

6.1 Functions of Various Variables
6.2 Limits and Continuity
6.3 Partial Derivatives
6.4 The Chain Rule
6.5 Gradient
6.6 Maximum and Minimum Values
6.7 Lagrange Multipliers

7. Multiple Integrals

7.1 Double integers in rectangular coordinates
7.2 Double integrals in polar coordinates
7.3 Triple integrals in rectangular coordinates
7.4 Triple integrals in cylindrical coordinates
7.5 Triple integrals in spherical coordinates
7.6 Jacobian of the coordinate transformation

8. Vector Calculus

8.1 Vector fields
8.2 Line integrals
8.3 Green's theorem
8.4 Rotational and Divergence
8.5 Stokes' theorem
8.6 Divergence Theorem



 

 

Learning activities

Presentation of the concepts in class, examples, solution and proposal of problems to solve.

Preparation in small groups of problems or topics to be developed previously discussed in class.

 

Evaluation system

There will be two exams during the course, a first partial and a second partial or final exam.

A collection of problems or practical topics to be developed will be proposed.

Those who have failed the first part will have to be examined in this part in the final exam, which will be a combination of the contents in the first part plus that of the second part. Those who fail the final exam will go for recovery. 

Each partial exam scores 45% of the final mark and the practices 10%.

REFERENCES

Basic

Notes of the subject

James Stewart. Calculation of a variable.

James Stewart. Calculation of several variables.